Equity Method Investments Investee Revenue

No definition available.

Name:	kg_EquityMethodInvestmentsInvesteeRevenue
Namespace Prefix:	kg_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

Gain (Loss) on Equity Method Investments

+ References

No definition available.

+ Details

Name:	kg_GainLossOnEquityMethodInvestments
Namespace Prefix:	kg_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

The percentage of ownership of common stock or equity participation in the investee accounted for under the equity method of accounting.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 825
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 28
-Subparagraph (f)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482907/825-10-50-28

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Subparagraph (a)(1)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481687/323-10-50-3

+ Details

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

- Definition

This item represents the carrying amount on the entity's balance sheet of its investment in common stock of an equity method investee. This is not an indicator of the fair value of the investment, rather it is the initial cost adjusted for the entity's share of earnings and losses of the investee, adjusted for any distributions (dividends) and other than temporary impairment (OTTI) losses recognized.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 280
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 32
-Subparagraph (ee)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482810/280-10-50-32

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 280
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 25
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482810/280-10-50-25

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 270
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 1
-Subparagraph (i)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482964/270-10-50-1

Reference 4: http://www.xbrl.org/2003/role/disclosureRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481664/323-10-45-1

Reference 5: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Subparagraph (a)(3)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481687/323-10-50-3

Reference 6: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(10))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

+ Details

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

- Definition

Amount of income (loss) for proportionate share of equity method investee's income (loss).

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Section 45
-Paragraph 28
-Subparagraph (b)
-SubTopic 10
-Topic 230
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482740/230-10-45-28

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 280
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 32
-Subparagraph (ee)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482810/280-10-50-32

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 280
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 22
-Subparagraph (g)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482810/280-10-50-22

Reference 4: http://www.xbrl.org/2003/role/disclosureRef
-Topic 270
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 1
-Subparagraph (i)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482964/270-10-50-1

Reference 5: http://www.xbrl.org/2003/role/disclosureRef
-Topic 944
-SubTopic 220
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.7-04(10))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147477250/944-220-S99-1

Reference 6: http://www.xbrl.org/2003/role/disclosureRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481664/323-10-45-1

Reference 7: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 220
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 2
-Subparagraph (SX 210.5-03(12))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483621/220-10-S99-2

Reference 8: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 942
-SubTopic 220
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.9-04(13)(f))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478524/942-220-S99-1

+ Details

Name:	us-gaap_IncomeLossFromEquityMethodInvestments
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
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

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 205
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 7
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483499/205-20-50-7

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 205
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 5A
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483499/205-20-50-5A

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Subparagraph (c)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481687/323-10-50-3

Reference 4: http://www.xbrl.org/2003/role/disclosureRef
-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 5: http://www.xbrl.org/2003/role/disclosureRef
-Topic 825
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 28
-Subparagraph (f)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482907/825-10-50-28

Reference 6: http://www.xbrl.org/2003/role/disclosureRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.4-08(g)(1)(ii))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480678/235-10-S99-1

Reference 7: http://www.xbrl.org/2003/role/disclosureRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Subparagraph (a)(1)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481687/323-10-50-3

Reference 8: http://www.xbrl.org/2003/role/disclosureRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Subparagraph (a)(2)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481687/323-10-50-3

Reference 9: http://www.xbrl.org/2003/role/disclosureRef
-Topic 323
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481687/323-10-50-3

+ Details

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

- Details

Name:	srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis=kg_SilverstoneVenture1Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis=kg_SilverstoneVenture2Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis=kg_SilverstoneVenture3Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis=kg_ExtellHudsonWaterfrontHoldingsMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis=kg_SeidenLPSeidenMGMTLPMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

Investments - Schedule of Carrying Value and Beneficial Ownership Percentage of Equity Method Investments (Details) - USD ($) $ in Thousands	3 Months Ended			6 Months Ended
	Jun. 30, 2026	Mar. 31, 2026	Jun. 30, 2025	Jun. 30, 2026	Jun. 30, 2025	Dec. 31, 2025
Schedule of Equity Method Investments [Line Items]
Carrying Value	$ 110,283			$ 110,283		$ 112,443
Interest in income of equity method investments	10		$ 0	9	$ 0
Realized and unrealized gains (losses)				(1,300)
Silverstone Venture 1
Schedule of Equity Method Investments [Line Items]
Carrying Value	$ 943			$ 943
Beneficial Ownership	90.00%			90.00%
Investee Revenue		$ 3,933
Interest in income of equity method investments		1,484
Realized and unrealized gains (losses)				$ (1,366)
Silverstone Venture 2
Schedule of Equity Method Investments [Line Items]
Carrying Value	$ 4,342			$ 4,342
Beneficial Ownership	86.80%			86.80%
Investee Revenue		70
Interest in income of equity method investments		68
Realized and unrealized gains (losses)				$ (85)
Silverstone Venture 3
Schedule of Equity Method Investments [Line Items]
Carrying Value	$ 20,585			$ 20,585
Beneficial Ownership	70.20%			70.20%
Investee Revenue		85
Interest in income of equity method investments		60
Realized and unrealized gains (losses)				$ 142
Extell Hudson Waterfront Holdings
Schedule of Equity Method Investments [Line Items]
Carrying Value	$ 51,000			$ 51,000
Beneficial Ownership	25.00%			25.00%
Investee Revenue		440
Interest in income of equity method investments		(24)
Realized and unrealized gains (losses)				$ 0
Seiden LP & Seiden MGMT LP
Schedule of Equity Method Investments [Line Items]
Carrying Value	$ 33,413			$ 33,413
Beneficial Ownership	100.00%			100.00%
Investee Revenue		147
Interest in income of equity method investments		$ 10
Realized and unrealized gains (losses)				$ 9