No guaranty that this will work for you, also I am not responsible if you damage your keyboard.

Remove all key caps (store them somewhere in the correct order ;-) and carefully unscrew the marked screws.

Open the case and disconnect the USB wire.

Now it is a good time to clean everything :-D

When looking at the circuit board, it can be seen that the upper metal case is clamped to the board to ground pads. I measured the resistance between the case and the ground - no connection; there is simply paint on these clamps.

My easy solution - without the need of further disassembling the keyboard: 

1. scrap the paint from the upper side of the clamp.
2. solder a small wire to the circuit board ground pad below the clamp
3. solder the wire to the upper side of the clamp (use some flux)

I made this for three clamps (bottom left, bottom right and mid top).

Reconnect the USB wire. Measuring the resistance showed: the case and Ground/USB shielding is connected (as it should be).

Close the case and carefully insert the screws.

Now I hope that the problem with turned of LEDs is solved.