How about 'jikoni kwake'?
That's more likely to mean "in his kitchen", but I don't want to say anything definitive about that.
I would second @AGreatUserName. "anapika jikoni kwake" for "he is cooking in his kitchen" but if you're just talking about the place without putting anything "in" it, it'd be "jiko lake."
When is it appropriate to use "kw-"
Does it work when using the "-ni" locative suffix?