I took a quick look in the schematic as I happened to have it spread out anyway... It seems that this fault can't be too complicated if everything else works normally. It's either a short circuit on the board (dunno how that could happen spontaneously) or the MM5450 IC that controls the LEDs has a pin stuck low.
The chip, although being quite old, is still in production, so getting a new one should not be too much trouble. Not too bad physically either since it is located in the front panel, which is quite easy to disassemble. Of course, if the TP2 input has nothing connected, the simplest "fix" would be to disconnect the two LEDs that light it...