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Balance Sheet Components - Schedule of Prepaid Expenses and Other Current Assets (Details) - USD ($) $ in Thousands	Sep. 30, 2025	Dec. 31, 2024	Dec. 31, 2023
Balance Sheet Related Disclosures [Abstract]
Deferred contract costs		$ 0	$ 1,026
Prepaid expenses and other receivables	$ 4,516	1,610	978
Payroll tax credit receivable	400	650	900
Total prepaid expenses and other current assets	$ 4,916	$ 2,260	$ 2,904

X

- Definition

Payroll tax credit receivable.

+ References

No definition available.

+ Details

Name:	kdk_PayrollTaxCreditReceivable
Namespace Prefix:	kdk_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

X

- References

No definition available.

+ Details

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

X

- Definition

Sum of the carrying amounts as of the balance sheet date of deferred costs capitalized at the end of the reporting period that are expected to be charged against earnings within one year or the normal operating cycle, if longer.

+ References

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(8))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

+ Details

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

X

- Definition

Amount of asset related to consideration paid in advance for costs that provide economic benefits in future periods, and amount of other assets.

+ References

No definition available.

+ Details

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

X

- Definition

Amount of asset related to consideration paid in advance for costs that provide economic benefits in future periods, and amount of other assets that are expected to be realized or consumed within one year or the normal operating cycle, if longer.

+ References

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(9))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

+ Details

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