Take a flashlight to the Nano's display or use it in a dark room to see if you then see the display. If so, it's likely that the backlight has gone out, meaning you'll actually need to replace the entire LCD display.
If you don't consider yourself handy with repairing small electronics, you can send it to Apple to be serviced for flat fee or you can choose to go with a third party service center who can do the job for a little less.
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