Standard Midwest city fare, mostly. Maybe the most unusual thing about KC is the changes between rich, not rich, and poor neighborhoods and how quick it happens and how close they are to each other.
KC is REALLY spread out and the statistics are all screwy from the Greater KC Area being in three counties across two states. I'm not sure if you realized this, but Overland Park is a suburb of Kansas City and has about 175,000 people. You aren't going to want to run from there to Kansas City, lol.
If you like botanical gardens, Overland Park has one on the south edge at like 179th and Antioch. It's somewhere around 300 acres. Shawnee Mission Park is a 1250+ acre monstrosity full of trees and lakes and trails and lakes and crap. I think it's one of the largest parks in the US, but I might be making that up.
18th and Vine is a historic Jazz district and has a museum as well.
I like the Nelson art gallery which is in KCMO and go there a couple of times a year.
My favorite BBQ (I'm a rib guy) place in the city is actually called Oklahoma Joe's, and the original location is connected to a gas station in Kansas City, KS sorta between where Mission, Shawnee Mission, and Kansas City meet. Best ribs IMO in the KC metro area, and they always have a line of people waiting, sometimes out the door. But you can hit up like twenty more places in KC that very well might all be the best BBQ you've ever gotten a chance at. I am not a great judge of BBQ because I'm stupid spoiled living in the Greater KC Metro area my whole life.
Union Station, host of the St. Valentine's Day massacre (a byproduct of that event is the FBI), is supposedly cool and has exhibits, but the only time I've ever been was for my prom 6 1/2 years ago.
If you want a real KC tradition, hit a Royals game. Due to the All-star games, the Royals' schedule, and your schedule, it looks like that Friday night might be your only shot. Nothing is more KC than seeing a Royals game and drinking away the pain and shame of having one of the worst franchises in sports (not just MLB) in an absolutely beautiful stadium.
Anything else I can do, ideas, etc... just hit me up.