Once the mess, if present, is cleaned you can insert your replacement 8mmx8mm switch and solder the pins to the underside of the main PCB.
Once the soldering is complete then so is this repair. Simply reassemble your device and hit your power button and it should power up as if there was never a problem. If you can't keep the original consoles alive at least continue the great memories of your favorite 8-bit games. Happy Retro Gaming!