I recently bought two of these keyboards and the LED thing was driving me nuts. So, I tore it open and took a look at the board. It's simple enough, I just soldered a jumper between the 5V from the USB line over to the 5V for the LED breakout board. This shot shows exactly what you need to do, just be careful not to solder the pads together or you'll probably fry something.
EDIT: I should add that it does work, but the LED's can't be turned off at all. It's on my GF's PC and I don't know what it does when the PC is off, probably depends on the USB port.
It's been working for all of 15 minutes, so I can't guarantee that it won't cause problems down the road, so proceed at your own risk. However, I can guarantee that you'll void any warranty you have.