Hi there!
I always replace both when this is the case!
It's most likely that the top screen it's ribbon cable is damaged already and thus need an replacement.
The hinge is a known issue with the DS Lite. Replacing the shell should solve this. There is most likely a small crack in the shell.
Be aware that there are some Chinese shells over there which are pretty bad... I recommend to replace it for an original one, but this will be incredible hard to find. So, when you do need to buy a new Chinese shell, make sure you get the same color as the one you have now. You can fuse the parts. Just only replace the broken part and then use the original parts of the shell to make it complete. I have seen DS Lite motherboards dying because of the cheap Chinese shells. Maybe there are some more expensive better ones on the market.
Please feel free to ask if you have any more questions! Good luck!!