After a few saturations and wipes with paper toweling, I had removed most of the alkaline gunk. From here you can clean it with a baking soda paste, but for this, I just decided to scrape the contacts a bit.
Once it was cleaned up, I put in some old AA batteries and it powered right up! I dropped in my old 2 gig EyeFi Geo card ($20 from Woot many moons ago), and it worked! I recharged some high capacity AAs overnight and dropped them in. Then things started going badly...
It would power on, then immediately turn off. I tried my second set of rechargables, and the same thing happened. I re-cleaned the contacts, but that didn't seem to help. It was at this point that I thought I knew why it had been dropped in the recycle pile.
Then I realized that my AA rechargables put out 1.2v each, rather than the 1.5v of alkalines. The camera has a requirement printed on it of 3.15 volts. 1.2 plus 1.2 does not equal 3.15.... I found a fresh pair of alkalines and dropped them in, and sure enough, it works perfectly!
As a sidenote, I also set up my computer at home to host the EyeFi software, and it drops the downloaded pictures into a Dropbox directory. Now, all of the pictures taken with the camera are instantly shared when I'm home. I have no reason to manually sync the card physically now. Yay!