The percentage discount that share shares are purchased from the then current market price.

No definition available.

Name:	kai_EmployeeStockOwnershipPlanEsopDiscountToMarket
Namespace Prefix:	kai_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

The maximum period of restriction on the resale of the shares purchased under the employee stock purchase plan.

+ References

No definition available.

+ Details

Name:	kai_EmployeeStockPurchasePlanEsppPeriodOfRestrictionOnResaleMaximum
Namespace Prefix:	kai_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Share-based Compensation Arrangement by Share-based Payment Award, Shares Received Upon Vesting

+ References

No definition available.

+ Details

Name:	kai_SharebasedCompensationArrangementbySharebasedPaymentAwardSharesReceivedUponVesting
Namespace Prefix:	kai_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	duration

- 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

No definition available.

+ Details

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

- Definition

Amount of cost for defined contribution plan.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 715
-SubTopic 70
-Name Accounting Standards Codification
-Section 50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480794/715-70-50-1

+ Details

Name:	us-gaap_DefinedContributionPlanCostRecognized
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Amount of cost not yet recognized for nonvested award under share-based payment arrangement.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 718
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Subparagraph (i)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480429/718-10-50-2

+ Details

Name:	us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Weighted-average period over which cost not yet recognized is expected to be recognized for award under share-based payment arrangement, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.

+ References

+ Details

Name:	us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedPeriodForRecognition1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Period over which grantee's right to exercise award under share-based payment arrangement is no longer contingent on satisfaction of service or performance condition, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days. Includes, but is not limited to, combination of market, performance or service condition.

+ References

Reference 1: http://www.xbrl.org/2003/role/exampleRef
-Topic 718
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Subparagraph (a)(1)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480429/718-10-50-2

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

The number of grants made during the period on other than stock (or unit) option plans (for example, phantom stock or unit plan, stock or unit appreciation rights plan, performance target plan).

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriod
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	duration

- Definition

The weighted average fair value at grant date for nonvested equity-based awards issued during the period on other than stock (or unit) option plans (for example, phantom stock or unit plan, stock or unit appreciation rights plan, performance target plan).

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriodWeightedAverageGrantDateFairValue
Namespace Prefix:	us-gaap_
Data Type:	dtr-types:perShareItemType
Balance Type:	na
Period Type:	duration

- Definition

The number of non-vested equity-based payment instruments, excluding stock (or unit) options, that validly exist and are outstanding as of the balance sheet date.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 718
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Subparagraph (c)(2)(i)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480429/718-10-50-2

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 718
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Subparagraph (c)(2)(ii)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480429/718-10-50-2

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

Fair value of share-based awards for which the grantee gained the right by satisfying service and performance requirements, to receive or retain shares or units, other instruments, or cash.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 718
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Subparagraph (d)(2)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480429/718-10-50-2

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodTotalFairValue
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

The highest percentage of annual salary that an employee is permitted to utilize with respect to the plan.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480429/718-10-50-2

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardMaximumEmployeeSubscriptionRate
Namespace Prefix:	us-gaap_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	instant

- Definition

The difference between the maximum number of shares (or other type of equity) authorized for issuance under the plan (including the effects of amendments and adjustments), and the sum of: 1) the number of shares (or other type of equity) already issued upon exercise of options or other equity-based awards under the plan; and 2) shares (or other type of equity) reserved for issuance on granting of outstanding awards, net of cancellations and forfeitures, if applicable.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

Number of shares issued during the period as a result of an employee stock purchase plan.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-SubTopic 10
-Topic 505
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481112/505-10-50-2

Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(28))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 3: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(29))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 4: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 505
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.3-04)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480008/505-10-S99-1

+ Details

Name:	us-gaap_StockIssuedDuringPeriodSharesEmployeeStockPurchasePlans
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	duration

- Details

Name:	us-gaap_AwardTypeAxis=kai_RestrictedStockUnitsForNonEmployeeDirectorsRsusMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_AwardTypeAxis=kai_PerformanceBasedRestrictedStockUnitsMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_AwardTypeAxis=kai_TimeBasedRestrictedStockUnitsMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_AwardTypeAxis=us-gaap_RestrictedStockUnitsRSUMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_AwardTypeAxis=kai_EmployeeStockPurchasePlanAwardsMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_TitleOfIndividualAxis=srt_DirectorMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

Employee Benefit Plans - Additional Information (Details) - USD ($) $ / shares in Units, $ in Thousands	12 Months Ended
	Jan. 03, 2026	Dec. 28, 2024	Dec. 30, 2023
Defined Benefit Plans and Other Postretirement Benefit Plans Table Text Block [Line Items]
Number of shares available for grant (in shares)	252,350
Cost recognized, defined contribution plan	$ 7,437	$ 6,810	$ 5,607
Restricted stock units for non employee directors (RSUs)
Defined Benefit Plans and Other Postretirement Benefit Plans Table Text Block [Line Items]
Granted (in shares)	527	606	868
Unvested RSUs (in shares)	0
Performance-based restricted stock units
Defined Benefit Plans and Other Postretirement Benefit Plans Table Text Block [Line Items]
Shares received upon vesting (in shares)	1
Vesting period (years)	3 years
Compensation cost not yet recognized	$ 3,673
Compensation cost not yet recognized, recognition period	1 year 4 months 24 days
Time-based restricted stock units
Defined Benefit Plans and Other Postretirement Benefit Plans Table Text Block [Line Items]
Shares received upon vesting (in shares)	1
Compensation cost not yet recognized	$ 4,798
Compensation cost not yet recognized, recognition period	1 year 9 months 18 days
Restricted stock unit awards
Defined Benefit Plans and Other Postretirement Benefit Plans Table Text Block [Line Items]
Granted (in shares)	28,000
Unvested RSUs (in shares)	61,000	74,000
Weighted average grant date fair value, granted (in dollars per share)	$ 365.15	$ 314.33	$ 212.92
Fair value of awards vested in period	$ 9,888	$ 9,089	$ 7,834
Employee stock purchase plan awards
Defined Benefit Plans and Other Postretirement Benefit Plans Table Text Block [Line Items]
Number of shares available for grant (in shares)	52,883
Discount to market under the ESOP	15.00%
Maximum period of restriction on resale under ESOP	1 year
Maximum participation based on compensation percentage under ESOP	10.00%
Stock issued during period under ESOP (in shares)	10,204	8,818	10,627
Director \| Restricted stock units for non employee directors (RSUs)
Defined Benefit Plans and Other Postretirement Benefit Plans Table Text Block [Line Items]
Shares received upon vesting (in shares)	1