Several things available.
Slim line lightboxes with snap frames are on the market.. most are off the shelf, but for the size you would want it would have to be custome made.
Another option would be to make the frame and back for the box and use some T5 flouros.. can run 2 of these off 1 electronic ballast. Electronics are good in this instance as they self adjust to load and give off no heat.

OR, go to jaycar and buy some of their "module"type LED.. about $25 per metre.. not the brightest LED in the world, but certainly good for what you want them for.

Some opal acrylic over the front as a diffusor and you have your self a slim line light box..... BG is applied over the top of this