Hi Evan, I would highly recommend the YMCA Aquatic Center on International Drive. It's surrounded by hotels, restaurants and shops, and is one of the largest indoor aquatic complexes in the southeastern US. You can pay a guest fee to swim there. Here is the website: ymcacentralflorida.com/.../
The YMCA Aquatic Center is also the location of the Rowdy Gaines Classic, if you are ever interested in coming to Orlando to swim at one of the best Masters meets in the country! www.swimspire.com/.../
Have fun!
Whenever I travel I use this site to find pools:
http://www.swimmersguide.com/
It relies on user feedback for changes and can require some phone calls on your part for verification, but I've always found a workable solution wherever I've traveled.