I fixed my problem. I gave up on libfaac, and instead just added the following code: audio_codec: "aac", custom: "-strict experimental -q:v 5 -preset slow -g 30" to the arguments of encode_video for it to not to try to use libfaac.
Also, I had the wrong MinGW. I had the git version, but I needed MSys to run commands such as ./configure and make and make install.