Landlord/tenant laws question

I moved into a place in September and didn't sign a lease or a rental agreement. Landlord told me she would need 30 days notice before I moved out. I agreed.

Despite what I was promised, the place was dirty when I moved in (I cleaned it) and several problems were supposed to be fixed but never were (among them, the back door was not secured at all and someone could easily break in, there was a rodent problem, etc). After 2 1/2 weeks I'd had enough. I looked up the landlord/tenant laws for Louisiana, and they state that, without a lease, the rental term is assumed to be month by month, and in order to receive a deposit back, the renter must provide 10 days notice of moving out before the end of the month. I gave her 13 days notice, and the address to send my deposit.

Not surprisingly, she didn't send me back my depost and I don't see her doing that. So, since I didn't sign a lease, the deposit is legally mine, right, even though I verbally said I would give 30 days notice? Apartment was in better shape when I moved out, so no damages occurred.

Thanks for the help