The AC will help dry everything out, but it won’t help with the smoke. It only cycles/recycles air. It draws air in from outside blows it over the condenser coils and sends it back outside* and it the air that’s in the conditioned space is continually blown over the evap coils.
Now that I think about it, your best bet is probably just to get a box fan, put it in the window, blowing out and close the bathroom door (no towel under it, don’t bother with the exhaust fan). Anything that creeps up should go right out the window. Worked for us in college.
As for the stairs. This is clearly bugging you. Maybe the next time the landlord is over you could let him see all the fans and just mention to him there’s a lot of smells drifting up and let him take care of it. If you’re lucky he’ll be the type of landlord who won’t get your involved he’ll just go down there and say that HE was upstairs and noticed it on his own.
*Assuming there’s two hoses (hose within a hose, often times) going out the window).
Have you considered removing the vanity doors so that everything can dry out faster? You can put them back as soon as things are dry, and then temporarily cover the hole while you await the handyman’s repairs.
