I had a switch pretty much identical to that give trouble a few months back.
Anyhow the best assist is a soldering clamp like this.
It is best if you can support what you need to repair.
Now I use a desoldering iron, but you don't need one, it just makes it easier.
Get the solder joints hot, one at a time. When the solder flows, just pull the wires out with pliers. If the hole fill with solder, then use a little piece of solid copper wire, heat the solder in the hole and poke it though with the copper wire. Keep the wire moving to and fro until the solder cools. Then you will have a nice clean hole.
Now tin the wires you will solder back, not too much. Poke the wire through the hole, and then let the solder flow. Make sure you use enough heat that the solder flows freely otherwise you will get a dry joint. Make sure that you do not use excess solder and create a solder bridge between the joints.
It is not difficult.