I did it! It worked. I even got the wires for free. I went to my local hardware store, and they told me that they didn't sell that type of wire. They had some in the back, though, and just gave me a few pieces. Cut it to size, connected it with wire nuts, and voila! It works.
Yes, it was a little crowded inside the fan. I had to stuff everything in, but it seems to be fine.
Thanks for the tip about soldering. I know that's an option, but I've never needed to own a soldering iron before and probably won't need one any time soon. It looks like those cost about $40+, whereas the four wire nuts I bought only cost $0.40.