New York City or the rest of the state? Notice requirements are different. If she gave notice on or before her move-in date, even assuming she didn't have a valid reason to back out of the lease, you may have a claim to part or all of her fist month's rent but you owe her the return of her deposit.

Keep in mind, you're acting as a landlord. You have to follow your state's deposit laws and can potentially make things a lot more expensive for yourself if you fail to follow the law in relation to when and how you return a deposit.