For whatever the reason it's really hard to find why, only 13 miles of Kansas has Route 66 going though it. The best explanation I can find is that Route 66 like I-44 only went from Missouri to Oklahoma, but some lobbying by Kansas businessmen managed to get a bit of a Route 66 added through a corner of the state, which makes sense since a lot of Route 66 found its way into different towns because of local lobbying.
Route 66 as you can guess only goes through three towns in Kansas; Galena, Riverton and Baxter Springs. It hits these three towns by making a path that is essentially a 90 degree turn through the state. It comes in from the East to hit Galena and Riverton, and banks to the South after that to hit Baxter Springs and onward to Oklahoma.
There are very few sites to see in Kansas, but none the less there are some. Galena for instance has a small but kid friendly museum in an old railroad depot complete with a locomotive, caboose, a tank, and a few other interesting artifacts. Between Riverton, and Baxter Springs you will find and old 66 landmark known as the Rainbow Bridge, named this because of its arches.
Riverton and Baxter Springs both have interesting downtowns complete with historic buildings and cafe's so its a good place to stop and get breakfast or lunch. Also be sure to look for the rusty old tow truck outside of Baxter Springs that was the basis for Mater in the movie Cars.
Even though Route 66 only traverses 13 miles of Kansas don't sell this this state short. It is a historic part of the route and your trip won't be complete without going through it. Also don't think you will get down all 13 miles in a matter of a few minutes. All three towns are medium-small towns with active downtowns, and as you know Route 66 loves to put you through downtowns. So expect to get caught in traffic here and there, also as a warning look out for the old fashion overhead stop lights hanging from the center of the intersections in those downtowns if your not use to them they are easy to miss.