Take a flashlight and shine it on the nano's LCD display. Do you see any signs of life from the screen? There is a good chance that the backlight for the Nano has gone out and is in need of replacement, which will require you to replace the entire LCD. You can choose to have Apple swap out the Nano for a flat fee or you can send it to a third party repair center to have them swap out the bad part for a little less.
Here is Apple's pricing: http://www.apple.com/support/ipod/service/prices/
If you don't see anything, it might not also hurt to give each of the 5 R's of troubleshooting a try.