Because of the kids, we drive at night. It makes the trip much easier and shorter on them. There are precautions you must take in order to do it, for safety sake.
I ALWAYS take a long nap on departure day--and make sure I get plenty of rest for three days before hand. We leave a couple of hours before dinner. It doesn't matter who drives here, although I usually do simply because I know the short cut around heavy freeway rush hour traffic and DH does not. We stop for dinner somewhere. I mean get out of the car, stretch, eat, and be ready to set out as though we just started. We'll drive for a couple of hours, stop at a rest stop, get out stretch, run the kids, use the restroom, and are off again. The kids usually fall asleep very shortly after and DH goes to sleep with them (bring pillow & blanket for all but driver). I drive first because I'm more rested and will tire first--DH naps to build up his strength. When I tire, we switch drivers. We both always get out at gas stations and stretch--it helps keep you alert when it is your turn to drive. As one tires, we just keep switching drivers. And whoever isn't driving, is sleeping. And if we are both tired, we stop for the night, even if it is 4 am. And once the kids are awake, we stop and stretch and run the kids about every 2 hours. It works wonderfully for us. And we usually get to Disney by mid-morning and have a full day there; amazingly enough with everyone fully rested. Somewhere in FL, we do stop a reststop, brush teeth, wash faces, change clothes, reapply deo. in case our room isn't ready.
The scary thing is, when we drive, I actually see more accidents (probably due to more traffic on the road) during the day than at night.