Sounds like you put the LED in series with the switch
I can't get a drawing at the moment but:
1) take the +12v and put it into the switch.
2) On the other switch terminal you need 2 wires
3) The first wire will go to the amp so that the switch turns the amp on/off
4) The second wire you need to connect a current limiting resistor around 600-700 ohms will give you 15mA
5) To the other end of the resistor connect the positive side of the LED (the neg side usually has a leg shorter than the other or is marked by a flat spot on the side of it)
6) now connect the neg side of the LED to ground
This way the amp will always turn on even if the LED dies. I think you ran it in series and did as stated above, turned it into a fuse.