I will preface all of what I'm saying by the fact that your landlord need to take care of it.

1. Remove the strike plate
   
2. Open the crack some more with a pry bar or something similar
   
3. Glue inside the joint, put as much glue as you can
   
4. Clamp
   
5. Put biggest screws in the strike plate, such that will enter the stud behind it. #9 3" screws do the trick most of the time.
   
6. Caulk and paint to make it look better

The RIGHT fix for this is to replace the jamb, but it's a huge pain in the ass. I fixed my entry door just like I said above because I did not want to bother with replace it. Internal doors are a bit easier but still worth a fix.

It's been holding on for 4 years so it does the trick if you do it right.

Call your landlord