Mapquest indicates 20.5 hours - that seems about right. The route would be more Westernly than I'm used to taking. So I can't speak to that, but I can speak to driving - long distances - with a young child.
We have made the trip 3 times in the last three years. We have one child - 3 (turning 4) on our first trip and 5.5 on our last trip. She has never been a problem. We have a lap desk for her that fits over her car seat, so she can color, draw, read, puzzles, games etc. Fairly inexpensive too, I would say $20. We also have a portable DVD player - more expensive but worth it. And we also bring along plenty of Kids music - Disney of course - for the drive. Our journey is 14-15 hours, and we usually try to get the bulk of it out of the way the first day, 9-10 hours.
We have found that having our car at Disney is a major major plus. Not to knock Disney transportation. But not having to wait for a Disney bus at the end of the day is major bonus - at least for us. And if you want to travel somewhere offproperty it is possible this way. Plus you won't have to worry about dragging strollers on and off buses, you can put them in your own car - and if you stay on Disney property - parking is free. Also transportation between resorts becomes much much easier, if you are going to use any of the facilities/restaurants at the other resorts. We've traveled with friends, who have gone both ways no car- Disney bus to the parks/our car and drive to the parks, and all agree - having a car is a major advantage, especially with kids.
Have a great trip and have fun planning.