Should I do anything with this piece, where the starter button makes contact and where the yellow wire is soldered too? I am assuming that the metal square with wires sticking out is the old solder. Would this need to be removed or when I solder to it will it melt and assist in the process?
Under lab conditions (rather than a field repair), I would clean off and discard the old solder and wire remnant, and start afresh. Flux also aids in solder removal, as well. This assumes you have a ready supply of solder for replacement.
Take note that the break did not occur with any of the tinned copper. Tinned wire is stronger and/or less flexible than the bare strands, and that is why it broke there, at the point were bending is hindered.
A third hand is a very nice tool to have, btw.