When I move my stuff from my stuff from my parent house to my relatives (helping them fix the house and I get to live rent free), I was told after the fact that my parents have a mouse problem. Here is what we did
1. Do not disturb the area that they live in. Meaning do not clean the area. Once you clear the area, they will scatter and you won't know where their next nest will be. Usually it will be areas of clutter where they can hide and out of view. Anything you do not want the mouse to touch, put in hard plastic 50 gallon containers.
2. Buy lots of glue traps. We ended up with 32 glue traps. Put chocolate on the center. Wherever you saw the mice, you need to place three traps in the room along the wall.
3. Do not eat in the areas where you saw the mice. You can leave crumbs. Starve them to the chocolate.
4. Do not leave loose paper around the floor or anywhere the mice can get to. Mice can eat paper, plaster,etc if there is nothing else to feed on.
Be aware that they have litter every 3 months (I think). So if you don't catch the mother, she can pop out another litter for you to have fun.
We caught about 7 little mice and 1 medium size and 1 very large size. Afterwards, we have not seen them since. Nor any evidence of leftovers (mice poo is a clear sign). Once confirmed, we sanitized the area and made sure we clean the house so it is free of crumbs, loose paper, clothing etc... there is nothing for them to eat.
If there are any holes in the walls, I suggest sealing them up with metal grate and plaster. They can eat thru plaster and push thru metal but at least it will take them awhile before they can break thru. Usually, they try to find another way in unless they are desparate.
And if you landlord is not pet-friendly, he needs to hire a better exterminator for those DAMN MICE!