Are you sure the drain snake didn't go up the vent stack above the roof ?
Send a certified letter to the SOB slumlord that if immediate action is not taken, he is violation of your lease and that you will hire ap lumber and deduct his bill from the rent.
Also document, document, document all interactions with him/her about this and other problems you have had.
Check with your city and the State website concerning renters rights and what can be done to either move out with all deposits returned or how to put a boot in the slumlord's ass to get him motivated.