There is a Jack's Stack not far from Union Station in Kansas City. You MUST eat there! Try the chessy corn casserole as one of your sides. Also, right in Union Station is Pierponts - a great steak house. Remember, steak "grows" in the area and is the best in the country in Kansas City!
The Plaza for shopping and walking around is great. There is a great variety of resturants on the Plaza, all kinds and all prices.
For history, the WW1 Museum is across the street from Union Station. It is definately worth seeing. It's sponsored by the Smithsonian Institute. Highly recommended.
Another Smithsonian Museum that is highly recommended is the Steamboat Arabia Museum in the River Market area (not far from Union Station/Crown Center/Plaza). This is the most amazing museum. If you see nothing else, go see it. The Missouri river was the highway for steamboat trade in the 1850's, but hundreds of them sank on the way. This one was dug up in a farm field by a local family and all the contents were perfectly preserved. The steamboat was taking all the supplies needed for a new town in Iowa - nails, china, door knobs, hats, pickles, perfume, everything. You can see parts of the steamboat and even the skeleton of the only fatality, a donkey. It's most interesting and definatly unique.
In Independence, MO (an eastern suburb) is the Harry Truman Library and also his house. They always have interesting exhibits. It's a far eastern suburb though, you'd need a car. Read the book "Truman" by David MacCallum before you come to Kansas City for some great history of the political scene. Also in Independence is a small museum at the start of the trails west - the Sante Fe trail, Oregon Trail and California trail. You can still see wagon ruts in the ground.
The Nelson-Atkins Art Museum is 1st class. It's not far from the Plaza. The Kemper Museum of Modern Art is also close by the Plaza area.
The Negro Leagues Baseball Museum is supposed to be very interesting, but I haven't been there yet. It's also by 18th & Vine, historical jazz center.
The Power & Light District and the new Sprint Center always have something going on. There are a great variety of restaurants in the Power & Light district.
If you stay around Crown Center (by Union Station) or in the Plaza Area you will be close to most of the above. The Plaza, Crown Center, Union Station, Power & Light, River Market etc. are all on the Missouri side of the state line, kind of downtown, a great area to use as a base for your visit to KC.
Kansas City is sprawled out quite a ways on both sides of the state line. I would stay central (Plaza or Crown Center) to avoid a lot of car time.
Have a great trip to KC. It's one of the best kept secrets in the US!
(If you can stay longer, take a day trip thru the Flint Hills. It's uniquely Kansas. Imagine the settlers trudging with their covered wagons over the plains. And see the steaks "growing"!)
Also, St. Louis - the Hill - Italian food. Can't beat it!
PS I was in York a few years back. Beautiful city.