At that size it would probably be using a CR123 type battery. Two of them in series will light up extremely bright LEDs. Ones you can get like a surefire type body will last about 20 hours on a pair of those but are only bright enough for "seeing" and won't light up a small room for photography.
Of course you can go nuts with with a diy led driver that pushes an led module with more than 5 watts. Taking a glass plate and grinding into a makeshift lens and sandblasting will give you a diffused refractor that you can reverse for wide or narrow beam in it's most simple form or you can even make a rail with the lens in a housing so you can "zoom" the light angle. If you only want diffusion then cut out an acrylic piece and sandblast or hand sand it down.
I built something like this for a few different projects, it's not hard if you can source all the parts. A module that's less than 10 bucks can shoot out 300+ lumens which is more than most Surefire weapon lights...
The most economical power supply would be li-ion 18650 sized batteries. They look like AA size but slightly bigger; almost all laptop batteries are composed of a bunch of these stuck together in a slab.
Here's a sample diy light source
you could build- it will provide slightly less than 700 lumens at 6500k color temperature and 70 CRI quality and will probably fit in an enclosure the size of a Rubiks cube. Runtime is 8 hours This is similar to the box you described earlier; it will light up a room with no problem.
for example, this photo shows a flashlight with "only" 500 lumens, so now you know what the device is capable of.
part / description / qty / approx price / source
- 5 watt led module / with heatsink / two of these / ~ $14 / superbrightleds.com
- LG 2600mAh li-ion cells / 18650 size / six / ~ $45 / batteryjunction
- 1.0 ohm 1w resistors / vishay dale or milspec / two / ~ $3 / mouser
- diffuser / sanded or blasted acrylic, maybe glass / one / ? / hardware store
- enclosure and switch
here is roughly what it would look like:
link to diagram
There are various ways to adjust the lighting angle and intensity.
By having the leds inside a can on a "rail" system with a parabolic reflector, you can adjust between a wide angle light or a spot light.
this photo shows what I'm talking about.
There are three ways to adjust light intensity:
Use paper gobo or something = waste of power, easy, free, lower intensity
A switch to halve the voltage = more wiring, 7v led may not be efficient at 3.6v, lower intensity
Have four 5w leds instead of two, wired to turn on 1 or 2 or 4 at a time = jungle of wires, efficient you get three intensity settings:
350 lumens@ 20 hours
700 lumens@ 8 hours
1400 lumens@ 3.2 hours
Obviously the third is the best. It isn't very expensive because you are only adding two more leds; most of the money is in the battery and charger. Labor is another story :P
Recharge the batteries; CAUTION
You'll note that I haven't mentioned how to recharge the batteries. It depends on whether you want to use an external charger with battery slots to charge each 7.2v pair (very tedious) or will be keeping the batteries inside(convenient and safe) and using a "battery pack charger"(like a normal charger but doesn't hold batteries, it has wires instead that you have to lead to something to charge.)
Either way each pair of batteries will need a protection device attached to prevent dangerous overcharging and a damaging over-discharge when low on juice.
The safest way to charge the system would be like this
Wall -> this charger, set to 8.4v -> the light system's three jacks -> each jack to the respective battery pair's protection board
You need three dc jacks unless which is safest method, using switches or other random ways to save on wiring may result in explosion.
The whole kit with three power settings and best charging method will cost $100 ~ $150+ including random stuff like wires, enclosure and switches.
a word on safety
This whole contraption may explode if you are not not careful, or even randomly. The 15" MacBook Pro also has six cells and it runs for 7 hours so you can sort-of imagine how big the explosion will be, lol.