I would reset the iPod first
Press and hold the home and power button, and it will reset
If that fails, restore in DFU mode
To get into this mode, plug the cable into the computer
press the power and reset buttons till it resets
let go of the power button, but keep pressing the home button
If that fails, replace the battery
If that does not help, replace or reseat the screen connection in the iPod, which does mean you need a need a new display assembly if reseating is no help
here is that guide
iPod Touch 4th Generation Front Panel Replacement
If you want to replace the backlight coil, this is a plausable try, but it's tough, worse then the iPhone 3Gs due to the smaller size of the backlight coil
here is a coil
http://www.ebay.com/itm/iPod-Touch-4th-L...
if you decide not to replace the backlight coil, you can sell for parts due to the fact i always suggest this more often then not due to the cost of the logicboard if that either A) does not help, the logicboard is fried or B) you don't have the experience, or time to change, and use the as-is iPod money to offset the cost of a new one
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Hi zoie, were/are you able to charge it? If not, can you remember if the battery was running low? If you can, how long - when fully charged - can you use the iPod in one session?
by Steffen