Even if you didn't have a formal lease, if you both agreed in a series of emails to contractual terms, then you have met the requirements for an agreement in writing. The agreement was ratified when the woman came and gave you money. You have met your duty of attempting to mitigate damages by trying to find another renter. So, you should not have to refund the money.
Answered on Dec 19th, 2012 at 1:50 PM