Hi Diane,
Disney's Old Key West Resort has a laid-back Florida Keys theme, and I find that it is a great place to relax on vacation. It is a sprawling resort. I like the relaxed and community-like feel that this offers; however, it does mean that there is a lot of ground to cover as it is a very large resort. You mention that you have stayed at Disney's Saratoga Springs Resort & Spa, and I find similarities in how these two resorts are laid out. They both include many buildings spread out over a distance.
Disney's Old Key West Resort features a main pool located near the Hospitality House, as well as three quiet pools located in the Old Turtle Pond, South Point and Miller's Road sections of the resort. If you booked through Disney Vacation Club and chose the category of "Near Hospitality House," you will be close enough to the main pool to walk to it fairly quickly. If you booked your room through another means other than Disney Vacation Club, you can request a room close to the Hospitality House. If you are in another location around the resort, it could be a long walk to get to the main pool.
If you and your daughter will be happy heading to a quiet pool, there are buildings close enough to those to walk. There are also multiple bus stops throughout the resort, so if bus transportation is a possibility for you, you can take the bus to the bus stop at the front of the resort near the main pool and take the short walk to the main pool from there. You can request a room close to a bus stop or pool. I know that you are trying to avoid renting a car; however, if you decide to do so, know that I often do and have found Disney's Old Key West Resort quite easy to navigate by car.
Please do note that while Disney will do their best to honor requests, they cannot be guaranteed. Please let us know if you have any further questions, Diane!