After an hour and a half in the Apple store tonight, my computer is back up and running!
The battery was replaced, the cord, with a little bit of jiggling and moving around, is working again and they even fixed the remnants of last years computer disaster (after a particularly bad crash I reloaded all the start up disks, but every time I accidently pressed the tab button on Safari, the program would freeze and then crash... which was annoying because it would take quite a while to go through the same process). The guy helping me seemed pretty surprised when he went into the computer and saw how often it was crashing. After trying a couple things they reloaded the program and things are running fairly smoothly again.
I have a few things to catch up on after not having a computer for a couple days, but hopefully I'll have a post that talks about something other than my computer being fixed soon!