How to connect external switches to this circuit board
Here's how to solder it neatly:

As Polynomial suggested, First rub the gold pads with a little fine sandpaper (1200 grit), or a salmon Garyflex block. Clean it with some methylated spirits. Run a little hot solder onto the circular ring, and the spot in the middle, taking care not to bridge them.

Trim both ends of your wires to 1.5mm, then tin them. You may find that the insulation shrinks back a little. Twist the wires together. Lay the wires on the PCB so that the tinned ends sit nicely over the solder. Tape them in place. Add a tiny amount of flux to the solder. With tweezers, position one wire over the solder on the ring. While pressing it with your thumb, melt the solder, so that the tinned part of the wire sinks into the solder. Do the same thing for the wire going to the centre spot.

Test that the wires work by touching the bare ends together. Does this do the same as a button push?

Now, clean the solder joints with some flux cleaner and a toothbrush. And lastly mix up a little potting compound or other epoxy, and spread it over the solder joint and a little over the wires so that they are safely glued down to the PCB. Once the glue has set fully (be patient), you can attach the button. Don't forget to use some heat shrink tubing.

This will create a strong, reliable solder joint that should last a lifetime.