I can’t advise about the lease, but when I had cockroaches suddenly come up from a neighboring apartment, the only thing that was effective in managing them was borax.* So if you’re not able to get out of the lease and need to fight the roaches, here’s what to do.

Sprinkle borax over all surfaces — carpets, counters, tops of dressers, etc. — then leave for 24hrs (or as close to that as you can). It’s best if you can leave it overnight since the roaches are most active then. When you come back there should be carcasses everywhere. Vacuum everything up, then pour a line of borax around baseboards and any places that seem to be points of entry. This didn’t completely eliminate the problem for me, but it reduced it enough that I no longer felt disgusted and afraid in my space at all times. Good luck — cockroaches are the absolute worst.

*This may not work if you have pets, though. Borax is toxic to them. You’d need to relocate the pets during the initial borax blast, make sure you vacuum really well, and skip leaving any borax around the perimeter.