I have an lg LW1515ER with the constant running compressor. What i did to fix it (so far anyway) was traced the compressor wires back to the control board where i found a brown wire and a black wire attached to a relay- what i did was disconnect the brown wire (while unit was still plugged in but turned off - compressor still running) and then attempted to touch the contacts together (from lead to relay).
Be careful, shock risk, you'll see the sparks fly. After that i simply unplugged the unit and reinstalled the brown wire and plugged the unit back in. Now, the compressor cycles regularly and if you switch to fan (or adjust temp up to turn off the compressor) and back to cool - after 3 minutes the unit behaves as it should. So, try that and good luck.
I also agree about just replacing the relay. With a little electrical know how, there's no reason i can see that any 110v relay capable of carrying at least 15amps (mine draws 12amps running - maybe up to 18amps or so on start-up) though -didn't check) wouldn't work as long as the other circuits are complete.
From what i saw there are only the 2 wires (1in and 1 out) so its really as simple as that. If mine ever decides to quit again, thats what ill do. There's plenty of space in there to just disconnect the old and reconnect a new one without further breakdown of the control board or even attaching it to the control board permanently. Good luck
