Brian, this may appear to be silly to you but since I can not see your ipod I would like to start with the basics. Try the settings first.
You can configure iPod so the backlight turns on for a set amount of time when you press a button or move the scroll wheel.
To access the backlight settings:
1. Select Settings > Backlight Timer.
2. Select a setting from the menu. Options include having the backlight timer:
- Turn off after a period of time between 2 and 20 seconds
A backlight on/off switch can be added to the Main menu by selecting:
Settings > Main Menu > Backlight
Of course you can also try to reset your Nano by following the Apple instructions
If that does not work then you have either a failure on the backlight of the screen which would be resolved by replacing the LCD, or it could be that the logic board has issues with the circuit that drives the backlight. I am not sure if the 5th gen has a coil on the logic board that gave previous models problems with the backlight. So to recap, Try settings, then replace the screen and hope it works. If not you are looking to replace the logic board. Good Luck and let us know how it is going.
iPod nano (5th Gen) Color Display
iPod Nano 5th Generation Display Replacement