The kit should be enough to get you started but you'll need something to get rid of solder like a desoldering pump or wick. I would get a desoldering pump with a silicone tip similar to this (there's cheaper versions online). You're mostly dealing with through-holes with switches so a pump is more useful but a wick is always nice to help.

If you plan on doing more soldering I would get something to practice on, some flux, brass sponge and tip tinner to clean, and more wire of course. Using a wider tip helps with getting all of the switch pins.