YouTube 14.43.55 is generally recognized as the last official stable release compatible with KitKat.
If you have the app installed but it won't play videos, try these common fixes: Google Groups Download Youtube Apk Android 4.4.2 - Google Groups
Designed for low-end hardware, YouTube Go 1.29.53 (or variants up to 3.08.58) often runs smoother on older devices.