I am looking for a small free tool that will allow me to add some audio to my video. Windows movie maker doesn't seem to work well from some reason.
I just used Windows Live Movie Maker exactly for that. It could be that it differs from Windows Movie Maker (which was superseded, afaik) on this aspect. You should try it, if you can.
The command-line tool ffmpeg can do this perfectly well.
If the input video alread has an audio track that you want to replace, you have to use
It's an amazing tool, and if you want to you can do almost any editing task with it (short of chroma key, and even that will probably be added at some point).
See here for how to get ffmpeg (in your case, scroll down to 'Windows Builds').