There are a number of hotels, resorts and rental homes on Sanibel Island. I recommend:
The Parrot Nest: Very old Florida! This is a small resort, from 1955, with 6 rooms each with a kitchen. It is near restaurants, the lighthouse, shelling beaches and the fishing pier. I enjoyed my large room and the ability to walk to breakfast or the beach. 1237 Anhinga Way.
Their sister property, The Kona Kai Motel has a pool that guests at the Parrot Nest can use. 1539 Periwinkle Way.
Sundial Beach Resort & Spa: At 1451 Middle Gulf Drive, this is a nice resort on the Gulf side of the island. They feature studio apartments along with one, two and three bedroom units each with kitchen. Its Seabreeze cafe is a good spot. It also has Shima Japanese Steakhouse & Sushi Bar and a pizza restaurant and beach bar. Other activities include spa services, tennis, fitness and access to golf.