It sounds like the daughter sync’d the iPod to the desktop iTunes library, which had a set of 160 songs, then sync’d the iPod to the laptop iTunes library, which had a different set of 71 songs. Syncing an iPod to an iTunes library means that the iPod’s entire contents will be matched to the contents of that iTunes library. This is a problem, as you saw, when the libraries on the desktop and the laptop are different.
You will therefore need to copy the songs from one library to another. Since the laptop iTunes library is smaller, I suggest you copy those songs. There are a number of ways you can do this, but the most convenient is probably to use the iPod itself to copy the songs.
First, plug in the iPod to the laptop and allow iTunes to come up. In iTunes, select the iPod in the source list and click the Summary tab. In the Options section, select “Enable disk use”, then click “Apply”. The iPod should now appear as a new drive.
Second, copy the song files from the iTunes library to the iPod. You’ll have to figure out where the songs are actually stored, but (assuming Windows) it will be in someplace like “My Music\iTunes\iTunes Music”. You can simply drag the entire “iTunes Music” folder (note: not “iTunes”) to the iPod.
Third, click on “Eject” and disconnect the iPod from the laptop. Connect it to the desktop. Wait for iTunes to come up. Go to “My Computer” (assuming Windows) and open up the iPod drive. Look for the “iTunes Music” folder that you just copied to it. Now drag that folder onto iTunes. This should copy the music files into the iTunes library. This may take a while.
Fourth, select the iPod in iTunes. Click on the “Sync” button and wait for the iPod to synchronize.
Fifth, delete the “iTunes Music” folder from the iPod drive. You can’t do this from iTunes so you’ll have to go back to Explorer (assuming Windows) and drag it to the trash. Once that’s done you may also want to disable disk use on the iPod. This is just like step one except that you uncheck “Enable disk use”. On the other hand, you may want to sync the laptop iTunes library to the desktop iTunes library, in which case you’ll have to repeat the above procedure but from desktop to laptop.
To prevent future confusion, I strongly suggest that syncing the iPod be done on only one system, either desktop or laptop. Any new music should also only be added to that system.