I've built light boxes for negs- mine is huge- 4' wide by 20" deep. I built it out of pine wood sides, 3x dual 4' fluorescent tube fixtures, and some white 1/4" acrylic I picked up for free (should have been 30$) from a plastics distributor.
Depending on what you have to display and how much you want to show I'd recommend going online and finding a used one- folks are clearing out light tables like crazy since film is disappearing. If you don't want to, just build a box that'll fit your light tubes, add some ventilation holes, and put a diffuser in (just like you've described).
My lightbox will do nearly 15 rolls of film, uncut, so I can review everything at once. I doubt you need *that* much.
Oh yeah- it's very very very very bright
(Yes, I know, I saw you wanted to use LEDs... but you can't get appreciable diffusion and intensity that you need at that distance. But)
If you use glass it isn't portable- it's fragile. Using acrylic will fit- you can get a small piece from a cutting board rollup mat from BBB. Space the LEDs so that they have no hot spots- you can't get great diffusion any other way (or double diffuse but that'll eat even more light).
The box? Get a project box from RadioShack made out of aluminum so you can heatsink directly and place sinks on the back.
Battery powered- little inset unit that can be removed... I'd consider buying a couple of lithium camera batteries and charger from eBay and using them (7.4V, 1900mah) to power it.