Change the battery.
I had the same problem with mine, I removed battery (its a 12v battery, V23GA) and tested it on the multimeter and read 9 volts. That was enough to stop the fob locking/unlocking the car but enough to light up the LED when pressing it. New battery and it was sorted. WHSmiths do the A23 battery which was about 1mm too short, so I just stretched the contacts up in the fob and it fitted nicely, not sure what type of battery yours uses though...Open it up and have a look.