I do believe that the 2nd generation iPod has a backlight coil on the logic board that has a tendency of corroding within the coil and thus shortening out. I do not of hand know which coil it is, but I know that is is difficult for DIY'er to replace the coil. SO I guess you would be looking at a logic board replacement. But I think the first thing you want to do is to dis assemble your iPod again and to make sure that all cables are connected right and tight. Also make sure that the cables are not fryed or broken, a good magnifier will help you with that. Clean the logic board and the connector with some isopropyl alcohol and do a visula inspection of the board. Reassemble the iPod and reset it and see what happens. Do pay close attention to any debris, corrosion etc. Good Luck to you and let us know if you are getting it accomplished.
iPod Nano 2nd Generation Repair