I usually use Booking.com's mobile app. If you look at rooms on your phone, they usually have some mobile discounts. Here are some options I see in mid-May from Booking using dates May 10 - May 31. (I didn't add tax rates, so prices will be more expensive than these after tax).
Habitacion cerca del centro - $576 for Double room with shared bathroom
- $645 for double room with shower
Pension Alicante - $641 for double room with shower and sink, shared toilet
Ruzafa Lander - $886 (plus $100 cleaning fee) for Studio apartment, private bathroom
- $960 (plus $100 cleaning fee) for one-bedroom apartment, private bathroom, small kitchen, washer machine.
Sundos Feria Valencia - $924 for double room with private bathroom.
I'm not sure how you plan on getting around Valencia, but public transportation is kind of shit there compared to other places like Madrid and Barcelona. So you might want to get a place close to your doctor, unless you have a plan of getting around. Never took a taxi there and I don't think they have uber or anything.