@shoileygal
Your pup is adorable.
For a house with fenced backyard I suggest looking at http://www.airbnb.com or http://www.vrbo.com. If go into filters, you can select "Pet Friendly"

The previous poster stayed at Homewood Suites by Hilton Rochester. It was across street from St Marys and husband was able to go back and forth from hospital and hotel to take care of dog. It has small fenced in patio in back of hotel.

This discussion talks about various places that accept pets:
https://connect.mayoclinic.org/discussion/best-pet-friendly-hotels/
When is your visit?