Steve Logan OSG
You need to be able to isolate the issue. It's either a failure on the backlight of the screen necessitating a screen replacement, OR it is a failure of the circuit that powers the screen's backlight on the main circuit board. If it's the latter, you may need a new mainboard. My best guess is the latter. If it was mine, I'd likely use it as is, as repair may be more trouble than it's worth. However, if time and parts money are no object, a screen replace has a chance of working, but a less than 50% chance by my estimation.
I've repaired hundreds of iPods. This is however, my guess, and that's all it can be from your description.
I had same issue with my 5th gen nano after it got washed. Was able to get unit functioning by placing it into uncooked instant rice for 3-4 days. That was about a year ago. Backlight has never functioned. I bought another unit (used) from apple. Kept it put up away from any chance of it getting washed. I have been using the unit with the none functioning backlight and turned on spoken menus option. I have been happy that I have been able to use the unit as it is. However, lately I have noticed that the headphone jack had not been seating full (snapping into position). I spent some time today looking with a light down in the earphone port. I could see something in the bottom. I took a paper clip, bent a leg of it straight, and carefully broke up the foreign matter. After a few rounds of tapping the unit several times on a semi-soft surface and picking out the pocket lint as well as some other unidentifiable matter, I was able to clear the port. When I plug my headset up, I was surprise to see the display come to life and function normally. It appears that when it got washed and dried out, that something was interfering with the triggering mechanism in the earphone port. Not sure why this effected the backlighting as it did.