To fix a door that won't latch, simply file the edge of the strike plate until the latch clicks into place. It's a five-minute fix.
Insert either a triangular file or a flat file that's small enough to fit inside the strike plate. File off enough metal to allow the latch to catch inside the plate and secure the door. If your filing action is shaking the strike plate, remove the plate and secure it in a vise for your repair. Then reinstall the plate with longer screws.
You shut the door. It doesn't latch, or you have to lift the handle to get it to latch. This door problem is getting on your nerves—why not spend five minutes to fix it? First try tightening the screws in both the handle and the strike plate. If the door still doesn't catch, grab a flashlight and study the way the door latch (the flexing part operated by the door handle) hits the strike plate. If the latch won't spring into the strike plate (Photo 1), observe whether it hits the plate at the top or bottom. If the latch seems properly centered on the plate, perhaps it's sticking on either the front or the back edge of the plate. When you discover where the latch is getting hung up, file that edge.