For the short term, you can by-pass the ignition switch, there are 4 wires on the switch, one is red that is 12 vdc+ from the battery, if you can connect a toggle type switch or any other switch that you may have, a 12 vdc rated switch is probably the better choice though.
Connect the red wire to one terminal of the switch, and then the other 3 to the other terminal in some jury rigged fashion, and when you turn the switch on, you should have a functioning bike.
You'll need to insulate the bare electrical connections, some heatshrink tubing, or at least electrical tape.
I would suggest removing the positive lead from the battery before attempting this, as the red wire is always 'hot', has power to it all of the time.