It has green, red, black and white wires coming out of it. I tried disconnecting the strips individually from each other and from the wires that were run to the other strips (I wasn't the one to put them in ) and it doesn't come on when I remove and reapply power to the box, so I assume the box itself has kicked the bucket.

The lack of a brand name kind of tells me right there it was a generic one, thus this was bound to happen sooner or later.

I don't really care to try and repair the existing box, what I'm looking for is some way to determine the requirements of the LED strips themselves when they, too, are unbranded. (Again, I wouldn't have chosen unbranded, but I'm dealing with this for someone else)

Unfortunately we don't have an analog variable power supply at our disposal, we only have ones with switches to select between 3, 5, 9 and 12 volts output.

 

... I assume the box itself has kicked the bucket.

You may be right but a failed LED can be enough to bring down a string of one color. It depends on how they're wired. The 6A rating makes me think the LEDs may be mostly in parallel.

Do you want full function restored, or just to light all the LEDs? I think you're out of luck for the former, meaning you'd be better off to just replace it all.

 

Do you want full function restored, or just to light all the LEDs? I think you're out of luck for the former, meaning you'd be better off to just replace it all.

If there was a problem with a specific strip, or just one color, I would assume the remaining ones would have come on when I powered it with that strip disconnected from the box. Yes I removed power from the box and reapplied it with a strip disconnected, and with just one wire disconnected, and nothing changed.

All we (the client) needs is just to be able to switch it to a static color and leave it that way. But to power the strip at all, I would need some method of determining the design of it (i.e. whether it has it's own resistors or not, and how to determine it's max continuous current non-destructively)

 

All we (the client) needs is just to be able to switch it to a static color and leave it that way.

Assuming the LEDs are good, I think you can get there with some experimentation. A variable power supply would be very handy for this but you can get by without it.

LEDs will make some light with even a very low current. This means you can apply a "high" (12-15V) voltage through a resistor, maybe 470Ω or more, and find out what voltage across the LED string is required to produce some light. The rest of the voltage will appear across the resistor. You can then reduce that resistance in increments to allow the current to increase. You can measure current by applying Ohm's law to the voltage across the resistor. Brightness is nearly proportional to current up to the point where the LEDs overheat and self-destruct. So make them only as bright as they need to be.