Part of the renovation of Kauffman included a kids area in the outfield, including a carousel and some baseball related games. Not sure where the best place for kids to sit are, but be aware that its a very sunny stadium. Bring quarts of the SPF. It is also probably the cleanest ballpark I've ever been to. I don't know what the concourse floors are made with, but they absolutely shine.
Kauffman isn't very close to the airport, but then again Kansas City isn't all that close to the airport. You're going to have to rent a car, so options of where to stay are pretty open. I know there are a few hotels near the Stadium complex, but you're in the middle of nowhere. We stayed in an area of he city called Country Club Plaza, which has plenty of shopping, restaurants, bars, and on street entertainment. It's still a bit of a hike to the park, but its a nice area.
2014 Home Attendance Record 3-10.