We drive to the parks (we have a rental car). However, since I was asked to overtake this thread, we tried a bus last year for first hand experience. We got on at Bus Stop 1 (the last stop picked up); we waited five minutes for a bus and took the bus to the MK. We had a seat and it took exactly ten minutes to get to the MK. I have read mostly good reports of the bus service (it helps that it is a convention resort and those guests aren't on the buses so the buses tend to not get filled.
A King room is a "bookable" category. So unless you booked a king room (which you couldn't do with 2 kids), you won't get one.