Yeah, could be tough getting even lighting. LEDs are spot sources so you'll need some diffusers and some depth (4+ inches) to the box to get the light to the center.
Personally I think just a cool frame that befits that scene and then go for a typical art-work light source mounted on the ceiling. They are low profile and can blend in well. Of course I'd say no way to the battery option as you'll hate them after a few changes. If you have access above (attic), running a low voltage power cord would be easy enough and hidden.