I think the input jack is the wrong type (it is supposed to be a stereo jack) or it is wired wrong (the tip is supposed to be audio, the ring to the negative terminal of the battery and the sleeve to ground).
The mono plug from the guitar is what connects the battery to ground because its long sleeve shorts the ring to ground in the jack. Without the mono plug, the battery is supposed to be disconnected then the LED will not light.
Hum indicates that a connector is wired wrong.