Answer

At first glance, the medieval authorities (Rishonim) disagree about this. The Darkhei Moshe (334:1) cites the Mordechai (Bava Metzia, no. 345) in the name of Maharam, that if a widow rented a house and died during the rental period, her heirs do not have to pay rent until the end of the rental term. But Rashba (responsa 1, no. 1028) writes that the renter’s heirs must pay until the end of the term, and his reason is this: “Since שמעון rented the house, it is as though it were an outright sale, and he became obligated for the payment whether he lives there or does not live there. And even though we hold that rental applies from beginning to end, nevertheless the renter may not leave every single day and calculate by the hour with him. Rather, he is obligated not to leave his house before the end of the term.”
In simple terms, the dispute depends on whether rental is a kind of sale or payment for use. If it is a sale, then certainly one must pay the whole amount even if one did not use it. And if it is payment for use, then there is room for the claim that half the money should be returned.
Since there is doubt here, the tenant is the one seeking to extract money from you, so the burden of proof is on her.
However, in your case, since she already paid and the question is whether to return the money to her, the Kehillot Yaakov on Bava Metzia, no. 37, wrote that according to all opinions rental is a sale and not payment for use. The entire dispute over whether rental is a sale or not applies only when the rent is paid at the end.
Therefore it seems that you are not obligated to return the money to her. That is my humble opinion.