IF YOUR NANO IS UNDER WARRANTY CONTACT APPLE.
If the failure is cause by water intrusion the best course may be to sell it on eBay AS-IS.
If none of above apply, you may consider a screen replacement.
Several generations of Nanos, all seem to suffer from LCD failure, "White screen of death"
This can sometimes be fixed by an LCD screen replacement.
Fortunately, on the 4th gen Nano, this is a relatively simple, but should only be attempted by an experienced technician.
See Fixit guide on this website
Replacement screens may be purchased on eBay for ~$10.