You should only need to replace the display assembly. Its a pretty cheap and easy fix on a 5S.
While you have the phone open, I would take a look at the battery, battery connector and any of the board you can see. Clean up whatever corrosion you see and inspect the battery for damage. If it was a quick drop, you might be ok.
If you still have issues after you replace the screen, then it will be time for some board level stuff. Especially watch out for quick draining battery, phone heating, etc.