I would try using a multimeter to check continuity when the switch is pushed in (the "door attached" position). This would be more safely done with the power off.
Start by touching one probe to the top wire and then touching the other probe to every other wire individually. When it beeps, those two wires should be connected together.
After checking the first wire, do the same thing for each wire and hopefully you will end up with two pairs of wires that have continuity. With the power off, connect each pair of wires together.
You can also apply the same technique in voltage mode to see if there is any residual electricity before cutting the wires.
Or you could just secure the button closed with crazy glue
