Unlike heat you probably cannot get the local authorities involved if there are windows to open and such. That said there may be an issue with not having all the equipment in working order as it was when you entered the rental agreement. Things get kind of technical and depending on how big a deal this to you'd need to talk to local attorney as to the nature of your lease...
Not clear if the place has multiple window units that are not getting the place cool or actually non-functional. It might make more sense to kindly ask the landlord if they can get at least one newer window unit to help cool the place, that is often cheaper to replace than repair...
You might even consider buying your own portable cooling unit --
https://www.walmart.com/ip/Shinco-YP...&wl13=&veh=sem I have seen folks sell these on Craigslist for next to nothing when they are moving and folks who sell them for top dollar during a heatwave. Timing is everything, maybe you can "make the arbitrage play" -- buy low, sell high...