Our unit has water damage from apartment above ours has

If there is a health hazard, like mold, you could request to terminate the lease. Given that you only have one month left, you could request the early termination as a concession as well, since it is almost over and you have to live with the damage.

In order to cancel the lease, if they won't agree, you'd have to argue this has made the apartment uninhabitable or is a breach of your lease. Is there a maintenance notice provision?

Legally, unless the fix would be hazardous with you there, or be so long and disruptive as to breach your right to quiet enjoyment they have the right to fix it with you in it.

Not if it were for example only one day. It would need to be more than a few days of inconvenience.

It would need to interrupt your use of the apartment for several days. The leak wasn't caused by their failure to maintain is it?

If they are the cause of the leak or their agents, you could use that, since their negligence is causing you these inconveniences. Terminating early could be a concession there. Was anything of yours damaged?

You should take photos of everything, especially look for mold from the damage. The issue is, if they don't agree to terminate and you just leave without letting them know you're leaving and their agreeing, they can try and sue for back rent for the final month. You should also gather anything they've said about the leak. If they've taken responsibility, that helps your case.

If they won't you could sue in small claims to cancel, based on their being the cause of the damage and the level of inconvenience.

Verbal testimony. You can document interruptions with your work. If you're in a meeting, you could have colleagues provide statements etc as to what occurred. You also can request notice of when the repairs will occur length etc, and document how often you have to request it and their lack of response.

It's between 64.50 and 86.50 depending on the amount of your claim. https://www.courts.phila.gov/pdf/brochures/mc/small-claims-pamphlet.pdf

You'd want to give them some notice and try and negotiate first, so they can't say you just moved out and never told them of the issues.

Trying to work things out to everyone's agreement shows you're acting in good faith if you have to go to court.

I understand, with that many issues, you can definitely claim they have been in breach of the lease, since maintenance is supposed to be provided quickly. Not being able to open windows is also a potential safety issue. How would you get out or firemen get in?

You're welcome. Right now, documenting everything is your best friend. You also can check with the city, since basement apartments are often not up to code. If it can't be rented legally, they can't hold you to the lease.