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I'm Lucy, and I'd be happy to answer your questions today.
A landlord has no ability to ask for additional deposits or cleaning fees after a tenant has moved into the premises, except when they are raising the rent. You do not have to give her a cleaning fee now, and she cannot evict you for refusing to provide it. It would take her more than 3 days to evict you, though, so if you're leaving at the end of September, she can't legally do anything if you refuse to pay.
Your obligation as a tenant is to leave the premises as clean as it was when you found it. That typically means it needs to be "broom clean" (empty and swept). You do not have to pay a cleaning company. If she would like to pay professional cleaners and you did NOT leave the place messy, then that money comes out of her own pocket. She can't charge you for it. Make sure you have pictures of everything when you leave, preferably with a copy of that day's newspaper to show that the apartment was clean on the day you left. Video can also be helpful (again, with a copy of the newspaper).
A person never has a legal right to damage someone else's property, so the fact that you don't have a written lease doesn't make a difference. The security deposit isn't a cap on damages unless you had something in writing that said it was (but written leases never say that). If you broke the doorknob, yes, she can charge you to fix it. If it's always been shaky, or if it's gotten worse through normal wear and tear, she can't. But she does have to prove that you broke anything she wants to charge you to repair.