Andrea, if you are sure that you booked a refundable room, spend the $100-200 (if even that) to have computer person look at your hard drive. This will give you the proof that you need for the hotel to refund your money.
I don't know the hotel system or how easy/hard it would be for them to be able to prove up the reservation at this point. Since you are the one asking them for either the favor, or for them to honor the agreed upon reservation, it is on you to come up with the paper work. I know, the hotel should be able to provide the reservation details as well, but they are not the ones wanting the money back, you are.
Have you offered to let the hotel keep the first night and refund you the rest, at least this would give the hotel something to work with?
One other question I have, were you charged in April or in July? This makes a difference to me, as generally, non refundable rates are charged at the time they are booked and fully changeable rates are not charged until the actual hotel stay. Of course, this could differ at different hotels.