Just the two switches damaged with no desoldering tools? Very impressive, unless they were the first two you tried before giving up :))
Try this (you can go in a straight line across the bottom three pins, I've just traced the paths as on the PCB)
Show Image
(http://i.imgur.com/90v1Wcu.jpg)
This worked for those keys! I tried using what method you seemingly used for following the trace in this photo for another key that no longer works (the spacebar key, to be exact,) and it didn't work. Could you give me some pointers on how to follow the trace like you did in that picture?
Thanks to everyone for being so helpful so far, you guys are a lot more friendlier towards someone learning than people on the mechanicalkeyboards subreddit were.
Probably not wrong, you just need more fixes. I expect you connected the spacebar to the diode (connection through the diode dotted as an example what's going on) but the signal would be cut where there's a break in the trace running along the bottom so it didn't work. Can't see where that trace goes, just hoping it's not straight to the pin on an SMD controller chip...
There are three red circles (:))) - one I can't see what's going on for solder, one chance of slicing through all the traces and a burnt switch out of focus - if you still have problems feel free to post pics of these areas, with the solder removed on the middle one.
Show Image
(http://i.imgur.com/M7D1j9G.jpg)
Bit of an update: I managed to get most of the keys working, now. I had to jump some other keys, but they're working! The ones that you circled in red were in rough shape, but most of them didn't need to be jumped. And it seemed that trace you circled in red was just some dog hair, I just checked the PCB and that thing isn't there anymore.
As I post this, I'm still trying to figure out how to fix the spacebar, as it's the only necessary key that is no longer functioning at this point.
Edit: I have managed to successfully jump the trace for the spacebar! I'm now working on finishing the rest of the keys. I will update this post if I need further help. If not, I will request that the thread be closed.
Edit 2: I have successfully jumped all of the traces and got it fixed. Thank you to everyone that helped me in completing my first actual trace repair. Mods, you can lock this thread now.