it sounds like the lower display has suffered enough shock that it broke but will occasionally display information. the good news is that the lower display is much easier to get to and swap out (you don't have to dis-assemble the entire DSlite).
replacement lower displays go for about 10-14usd, however you will need to transfer over your existing touch screen and the [black] double-sided adhesive frame the lies between the display and the touch screen when swapping out the lower display.
make sure that you carefully unclip both of the ribbon connectors. the larger one opens one away, while the smaller one for the touch screen opens in the opposite direction. (see the guide here on gbatemp for detailed information)
to remove the touch screen and adhesive frame from the lower lcd panel, I usually carefully run my thumbnail between them (starting at a corner to separate the 2 pieces (99% of the time, the adhesive frame will remain stuck to the underside of the touch screen vs. the clean, smoother metal frame surface of the lcd display) and then slowly work my thumbnail along one of the edges, separating the 2 pieces more and more until you can pull them apart.
make sure to thoroughly clean the surface of the lower display prior to re-assembly. for proper placement of the touch screen, I will usually drop it into position inside the lower half of the shell (so that it sits properly, blow some compressed air on the back side (to eliminate any dust particles than may have settled), and then drop the lower display on top of the touch screen. once you've done that, you can gently press down on the backside of the lower display so that the adhesive frame forms a seal all the way around the display, attaching the touch screen and locking out any dust and/or dirt from getting trapped between the 2 pieces.
after that, its just a matter of re-assembly in reverse order (again, using the guide). you want to be careful re-inserting both of the ribbon cables and also minding the positions of the plastics switches for both the on/off and volume controls so that they are in the same position as the actual physical switches so that you don't break anything when replacing the bottom side of the lower shell.
feel free to shoot me a pm if you have any questions or concerns