I had an old ds lite once, and it played fine, then one day it wouldn't turn on. the battery was good and everything.
I can almost 100% guarantee that you have either a lose wire, or a broken wire inside your ds lite. call around to diff places that can fix game systems, but it probly would be better to just buy a new one altogether at gamestop,, they are $50 bucks there REFURBISHED, which means new screen, battery, and they have soldered any wires inside that need it