We went to Weeki Watchie state park. That's probably not the closest place to Disney but it isn't terribly far, about 2 hours' drive, well within day-trip distance, and it was a fabulous place to see them. The launch is in the state park but outside the gates (shares a parking lot) and the river winds through both state park and residential areas. The manatee viewing is near the end of the 3ish mile route where there's a deep spring where they congregate and they come right up to "play" with the kayaks. It was about 40° the day we went so my girls were a bit nervous about the warning we got from the woman at the launch that sometimes they'll get playful enough to tip a kayak, but they didn't seem to want to spend enough time near the cold air to do that while we were there. At the end of the route, there's a park where you could swim too, assuming better weather than we had of course.
I absolutely plan to go back there again on our next driving trip down. It was an incredible experience that would have been even better in nicer weather!