jaj there are two distinct possibilities. One, it is a failure on the backlight of the screen, which would be repaired by a screen replacement, iPod Nano 5th Generation Display Replacement
Two, it is a failure of the circuit that powers the screen's backlight on the main circuit board. There are no schematics for this iPod (or any other iPod really). You could check for failed components and hope it's the backlight coil. Or you could replace the logic board (as above PITA) iPod Nano 5th Generation Logic Board Assembly Replacement The failed backlight is actually a pretty common issue with the 5th Gen Nano.