why not jsut turn your lappy off. put some probes on 2 legs at a time and then press the button and see what links.
Nomally on momentary switches (the little tactile ones) you get 4 contacts which are set in pairs. You switch might have a common connector eg middle and one pair of normally open and one pair of normally closed. Will take you a few minutes of trial and error to work out how it all works seeing what is connected to what and when.
As for the realy problem - no the led power would not be enough but you may beable to get a logic/led signal activated relay which has a circuit attached. Basically ithinkt his is just a transistor which applies the current ot the relay but i am b no means an electronics expert. The rela circuit you would need can be got from most little electronics hobby stores