I am sorry, but I do not understand some parts of your post.
If she gave her a notice to move out at the end of her lease on September 30, but she did not move out and she paid you rent for October, which you accepted and did not reject, and she moved out on October 31 (one month later), why are you indicating that she moved out "a week later than she gave notice?"
In any event, if you accepted rent from her after her notice, she is entitled to the return of her Security Deposit if she did not cause any damages to the Unit. Security Deposits and the procedures for Landlords to follow in order to retain part or all of the Security Deposits are governed by Texas Property Code Chapter 92 Sections 92.102 to 92.109.
Sec. 92.104. RETENTION OF SECURITY DEPOSIT; ACCOUNTING. (a) Before returning a security deposit, the landlord may deduct from the deposit damages and charges for which the tenant is legally liable under the lease or as a result of breaching the lease.
(b) The landlord may not retain any portion of a security deposit to cover normal wear and tear.
(c) If the landlord retains all or part of a security deposit under this section, the landlord shall give to the tenant the balance of the security deposit, if any, together with a written description and itemized list of all deductions. The landlord is not required to give the tenant a description and itemized list of deductions if:
(1) the tenant owes rent when he surrenders possession of the premises; and
(2) there is no controversy concerning the amount of rent owed.
Sec. 92.109. LIABILITY OF LANDLORD. (a) A landlord who in bad faith retains a security deposit in violation of this subchapter is liable for an amount equal to the sum of $100, three times the portion of the deposit wrongfully withheld, and the tenant's reasonable attorney's fees in a suit to recover the deposit.
(b) A landlord who in bad faith does not provide a written description and itemized list of damages and charges in violation of this subchapter:
(1) forfeits the right to withhold any portion of the security deposit or to bring suit against the tenant for damages to the premises; and
(2) is liable for the tenant's reasonable attorney's fees in a suit to recover the deposit.
(c) In an action brought by a tenant under this subchapter, the landlord has the burden of proving that the retention of any portion of the security deposit was reasonable.
(d) A landlord who fails either to return a security deposit or to provide a written description and itemization of deductions on or before the 30th day after the date the tenant surrenders possession is presumed to have acted in bad faith.