Hello, Mewtwodestroyer.
The button is a kind of 'micro switch' and can be found with that term in your searches on various online retailers. Examples can be found here on Amazon and here on eBay for your reference.
As for the matter of how you'd go about soldering with the necessary precision, learning the fundamentals of soldering is the first step if it's something you haven't done before. Here's a link to an iFixit guide by a contributor that may help. The particularities mean this isn't something that can be answered so briefly, but with the proper size of soldering iron tip, a good, clear view of the board, the correct equipment, and a steady hand, you should be able to find success.
crwdns2934105:0crwdne2934105:0
crwdns2934113:0crwdne2934113:0
crwdns2915270:0crwdne2915270:0
crwdns2889612:0crwdne2889612:0
2