A decent soldering iron. You will thank yourself later. Also note: you will burn yourself many times throughout the build. Might as well do it with a fancy iron!
LEADED SOLDER – I started the build without leaded solder and oh dear…
Well, even with a really crappy iron, as long as the tip is fresh, you shouldn't have a problem. Most people's problems with soldering come from not having the tip wet enough and not using enough solder, thereby not making enough contact with the wire and not putting heat into it fast enough to get that specific location hot before the heat conducts away into the rest of the wire.
Remember, the entire reason we're using the wire is because it's an excellent conductor. For that reason, thicker gauge wire is also going to be more difficult to work. And the only reason to use larger gauges of wire is to support more current. Keyboards run on extremely little current already. You probably could have done this with magnet wire and been fine (though I wouldn't recommend it, magnet wire is difficult to work for different reasons).
So crank up the heat. Get it into the wire as fast as possible. Lead-free solder generally needs higher heat than leaded solder anyway. Apply the solder, apply the wire, then wait a second before removing the iron, because you also have to heat the wire you're pushing in. Soldering is a chemical bond, the metals physically mix, and you need to get them up to the right heat to do that.
And keep it steady, any movement while it is cooling will create microfractures in the solder joint.