Mostly depends on the weather/current trail conditions. If it's dry, a road bike (with as wide a tire as will fit) would be better. When the trail gets wet, it can get soft - so a mountain bike would be better.
Not much of an answer, I know.
Another, shorter, option is the Wabash Trace trail (
http://wabashtrace.org/) - 62 miles in SW Iowa. Very nice, more shade than the Katy, and better camping options - many towns along the trail allow camping in city parks. As I recall, camping along the Katy is more tightly controlled; camping alongside the trail is not allowed.
