How to show iPod in iTunes

*How do we get the second computer to recognize her iPod?*

Have you had a chance to look at this troubleshooting page? It may be of some help: iPod shows up in Windows but not in iTunes

*we went back to the first computer to add the newly purchased song, but it doesn't show in that "purchased" list.*

That's because it was physically downloaded to the other computer, it's not on a list available to other PCs. If you want a song download on one computer to be available on another you have to copy the file over: iTunes for Windows: How to copy purchases between authorized computers

*I can't transfer a song from one computer to another if I can't get the song onto the iPod that isn't recognized by iTunes.*

There are various options in the link above for moving purchases that don't involve using an iPod, for instance:

Burn a data CD or DVD with the song files
Burn the song files to a data CD or DVD. Once you've burned the disc, insert it into another computer and import the songs into iTunes.

External hard drive
Drag files to the disk icon of an external hard drive connected to your computer. Connect the hard drive to the other computer, and import the songs into iTunes.