Thanks to this post here, I was able to figure out how to record Skype in Audacity. Although, the solution seems strange at this point. The result is hearing yourself a fraction of a second later in your headphones.
Per that post, you must have the "stereo mixer" installed. I had to install the Realtek driver to get this working right. From there, here's what I can figure...The Stereo Mixer, despite being listed in the recording devices section, actually seems to just take all audio sources in and mix them together. This means you can set Audacity to record from the Stereo Mixer and it'll get everything that's going on in the system. Unfortunately that also means that you hear your own voice just a fraction of a second later, like you're in a big hall and there's a hint of echo. It's a little annoying but it might still be usable.