You say that you have been advised by your lawyer as to how to handle the last month's rent, so why are you following up here instead of with your lawyer?
Florida Statutes, Sec. 83.03, describes the notice required to end a periodic tenancy: (a) An email is in writing and (b) you accepted and acknowledged the notice.
Your tenant does not appear to agree with you that she agreed to give sixty days notice. Do you have that agreement in writing, signed by your tenant? Do you have any confirmation from your tenant that she agreed to give sixty days notice to end her periodic tenancy, following the expiration of the initial lease term?
As for your having leased the premises as of September 1, your tenant has a point. You cannot simultaneously lease the premises to another person while claiming to hold her to her lease past the date you have committed the same premises to the new tenant. If the issue is that you have not finalized your agreement with the prospective tenants, but are nonetheless holding the property for them despite your having a significant concern that they will change their minds about renting, you should consider changing your business practices to better protect yourself -- for example, you can require a non-refundable holding deposit that will be applied to their rent or used as a security deposit if they move in (you'll want an appropriate written agreement to support your claims against any such deposit), or you can tell the prospective tenant that until the lease is signed there is a possibility that another tenant will come along who is willing to sign a lease in which case they'll lose their opportunity to rent.
Cleaning and painting will ordinarily fall under normal wear and tear, not damages that you could have deducted from a security deposit.
Although the difference is small, you are not prorating rent correctly. If rent is $1,200, and a month has thirty-one days, then you would prorate rent for the month by multiplying the monthly rent by the number of days for which rent is due, divided by the number of days in the month. So for August, with 31 days, each day would have a prorated rent of 1/31 times the monthly rent.
Does your lease provide that the tenant has to pay a "transfer fee" for the electric utility service?

