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.

Good Luck.