Unfortunately it could be a variety of problems. Fortunately they all have workable solutions.
Perhaps the easiest problem to fix would be if, when it was dropped, one of the cables became loose or disconnected. In that case all it would take to fix it would be to open the DS up and ensure all the cables are properly seated.
A more difficult one to address is it is possible that when it was dropped a component was knocked off the main board. And it is quite common for an inductor to get knocked loose in a drop, located near the power switch. But it is an easy solder if you happen to work with electronics.
If you want to further diagnose the issue I'd recommend using the following guides to disassemble the device and check for possible damage:
Nintendo DS Lite
Though there is another apparent option which centers around possible hinge damage. Does it appear that there was any damage to the hinge in the fall?
If so you may just need to replace the casing and one of the screens to rectify the issue. In which case the above guides would also be useful.
Also, if it turns out you need replacement parts, this is a good spot to start looking.