You could take your Nano to an Apple store for repair, depending on when it was purchased it may be covered by warranty, alternatively there are many iPod repair shops who advertise online. Try Google for examples.
If all the iPods are yours you can connect them to your library however you want. As the Nano may be be too small to load your whole library you may wish to set it to either "manually manage songs and playlists" or to "automatically update selected playlists only". The larger iPod you may want to simply set to "automatically update all songs and playlists". Have a look at this article:
How To Use Multiple iPods with One Computer
For info:
The iPod offers three ways to transfer music from your computer. You can select one of the following update modes from the iPod Preferences menu in iTunes (Edit=>Preferences=>'iPod' tab):
1) Automatically update all songs and playlists. This is the default mode, in which your entire music library, including playlists, is automatically synced to your iPod. If the music library on your computer exceeds the iPod storage capacity, you are prompted to select a different update method.
2) Automatically update selected playlists only. With this option, iTunes automatically copies the playlists you have selected to the iPod when you connect it to the computer.
3) Manually manage songs and playlists. You can also choose to transfer music to the iPod manually. This allows you to drag and drop individual songs and playlists from iTunes to the iPod.
