Change PCB-mounted switch
I can't find an explanation how the Button/Switch on the PCB works.
  Should I just try to solder the thing on there? Or would you recommend
  something else?


Get an ohm meter and find out which pins of the switch are connected to which. Then toggle the switch and ohm again. You might also be able to do this for the switch in the PCB while in circuit. If they match then great, you have a replacement.