Here's a pic of the other side of the board.
Wow, thanks for all of the great information and quick responses. I do in fact have a ps/2 cord, it is just disconnected from the PCB for now. I never even thought to use that for the power and ground. The PS/2 connectors are the five horizontal solder points toward the upper left corner of the first two pictures. The ground is the far left and the power is the far right of those 5!
As for my wires, I have discovered the hard way that they are not that good to work with. Thanks for the tip however. I am going to get some new wire in the coming days.
My projects so far have been mostly for learning purposes, so the wire I have now has been serving its purpose--but it really is a pain to work with.
As for melting the solder points, I thought that happened because I was just no good at desoldering. But, maybe it is because my iron is too hot (probably both). My iron is just a cheapo from Radio Shack and is marked as 23W. I'm also just using a little desoldering bulb to suck up the solder after heating it (looks something like the little suction bulbs for cleaning babies noses).
This mouse hack is going to connect to a Happ Trackball. I may also use it to try to build a spinner as well.
Again, thanks for all the help. You guys are great!