I changed the LEDs in one of my friends controllers a while back, and for some reason the 1st player LED always stayed on if there was a battery back in the controller, regardless if the controller was on or off, and regardless if the controller was 2nd, 3rd, or 4th player. Neither of us could figure out why it always stayed on hahaha.
As for your problem, I suppose check for any damaged traces or if anything is bridged, or try other rapidfire codes to isolate the problem.