Long-Term Obligations - Schedule of Long-Term Obligations (Details) - USD ($) $ in Thousands	Jul. 04, 2026	Jan. 03, 2026
Debt Instrument [Line Items]
Finance Leases, due 2026 to 2030	$ 2,393	$ 1,781
Total	510,632	374,501
Less: Current Maturities of Long-Term Obligations	(3,176)	(3,129)
Long-Term Obligations	507,456	371,372
Revolving Credit Facility, due 2030
Debt Instrument [Line Items]
Long-term debt	502,452	366,707
Senior Promissory Notes, due 2026 to 2028
Debt Instrument [Line Items]
Long-term debt	4,990	4,990
Other Borrowings, due 2026 to 2031
Debt Instrument [Line Items]
Other Borrowings, due 2026 to 2031	$ 797	$ 1,023

Amount of debt and lease obligation, classified as current.

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(20))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

+ Details

Name:	us-gaap_DebtCurrent
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 942
-SubTopic 235
-Name Accounting Standards Codification
-Section S99
-Paragraph 2
-Subparagraph (SX 210.9-06)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147477314/942-235-S99-2

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 505
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481112/505-10-50-3

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 942
-SubTopic 210
-Name Accounting Standards Codification
-Section S45
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147477657/942-210-S45-1

Reference 4: http://www.xbrl.org/2003/role/disclosureRef
-Topic 942
-SubTopic 210
-Name Accounting Standards Codification
-Section S50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478478/942-210-S50-1

Reference 5: http://www.xbrl.org/2003/role/disclosureRef
-Topic 942
-SubTopic 210
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.9-03(16))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478546/942-210-S99-1

Reference 6: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22)(b))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 7: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22)(a)(4))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 8: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22)(a)(3))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 9: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22)(a)(1))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 10: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S50
-Paragraph 9
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480657/210-10-S50-9

Reference 11: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22)(a)(5))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 12: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22)(a)(2))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 13: http://www.xbrl.org/2003/role/disclosureRef
-Topic 815
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 8A
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480434/815-10-50-8A

Reference 14: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480738/235-10-S50-1

Reference 15: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S50
-Paragraph 4
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480738/235-10-S50-4

Reference 16: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 3
-Subparagraph (SX 210.12-04(a))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480678/235-10-S99-3

Reference 17: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.4-08(f))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480678/235-10-S99-1

Reference 18: http://www.xbrl.org/2003/role/exampleRef
-Topic 835
-SubTopic 30
-Name Accounting Standards Codification
-Section 55
-Paragraph 8
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482949/835-30-55-8

Reference 19: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1B
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1B

Reference 20: http://www.xbrl.org/2003/role/exampleRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 55
-Paragraph 69B
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481568/470-20-55-69B

Reference 21: http://www.xbrl.org/2003/role/exampleRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 55
-Paragraph 69C
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481568/470-20-55-69C

Reference 22: http://www.xbrl.org/2003/role/exampleRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 55
-Paragraph 69E
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481568/470-20-55-69E

Reference 23: http://www.xbrl.org/2003/role/exampleRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 55
-Paragraph 69F
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481568/470-20-55-69F

Reference 24: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1B
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1B

Reference 25: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1B
-Subparagraph (c)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1B

Reference 26: http://www.xbrl.org/2003/role/exampleRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1B
-Subparagraph (d)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1B

Reference 27: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1B
-Subparagraph (e)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1B

Reference 28: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1B
-Subparagraph (f)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1B

Reference 29: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1B
-Subparagraph (h)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1B

Reference 30: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1D
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1D

Reference 31: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1D
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1D

Reference 32: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1D
-Subparagraph (c)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1D

Reference 33: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1E
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1E

Reference 34: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1E
-Subparagraph (c)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1E

Reference 35: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1E
-Subparagraph (d)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1E

Reference 36: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1F
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1F

Reference 37: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1F
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1F

Reference 38: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1F
-Subparagraph (b)(1)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1F

Reference 39: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1F
-Subparagraph (b)(2)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1F

Reference 40: http://www.xbrl.org/2003/role/disclosureRef
-Topic 470
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1I
-Subparagraph (c)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1I

Reference 41: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 835
-SubTopic 30
-Name Accounting Standards Codification
-Section 45
-Paragraph 2
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482925/835-30-45-2

Reference 42: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 835
-SubTopic 30
-Name Accounting Standards Codification
-Section 50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482900/835-30-50-1

+ Details

Name:	us-gaap_DebtInstrumentLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Present value of lessee's discounted obligation for lease payments from finance lease.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 842
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 6
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478964/842-20-50-6

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 842
-SubTopic 20
-Name Accounting Standards Codification
-Section 45
-Paragraph 1
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147479041/842-20-45-1

+ Details

Name:	us-gaap_FinanceLeaseLiability
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Amount, after deduction of unamortized premium (discount) and debt issuance cost, of long-term debt. Excludes lease obligation.

+ References

+ Details

Name:	us-gaap_LongTermDebt
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Amount of long-term debt and lease obligation, classified as noncurrent.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 942
-SubTopic 210
-Name Accounting Standards Codification
-Section S45
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147477657/942-210-S45-1

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 942
-SubTopic 210
-Name Accounting Standards Codification
-Section S50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478478/942-210-S50-1

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 942
-SubTopic 210
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.9-03(16))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478546/942-210-S99-1

Reference 4: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S50
-Paragraph 9
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480657/210-10-S50-9

Reference 5: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 6: http://www.xbrl.org/2003/role/disclosureRef
-Topic 944
-SubTopic 210
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.7-03(a)(16)(a)(2))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478777/944-210-S99-1

+ Details

Name:	us-gaap_LongTermDebtAndCapitalLeaseObligations
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Amount of long-term debt and lease obligation, including portion classified as current.

+ References

No definition available.

+ Details

Name:	us-gaap_LongTermDebtAndCapitalLeaseObligationsIncludingCurrentMaturities
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

The carrying amount as of the balance sheet date for the aggregate of other miscellaneous borrowings owed by the reporting entity.

+ References

No definition available.

+ Details

Name:	us-gaap_OtherBorrowings
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Details

Name:	us-gaap_LongtermDebtTypeAxis=us-gaap_LineOfCreditMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_LongtermDebtTypeAxis=us-gaap_SeniorNotesMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_LongtermDebtTypeAxis=kai_OtherBorrowingsMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: