I like the idea of recycling notebook backlights, rather than tossing them.

Building one from scratch is going to be pricey, because white LEDs have not
gotten really cheap. And all the ones I've seen have a bluish tinge which
might be disconcerting in the kitchen.

GE used to make "stick-up" fluorescents, which could be attached with screws
or double-sided tape. I don't think they're made anymore, but you might
look.