Unfortunately, although there are a ton of coffee houses in the downtown area, they're really not open late enough to be useful for studying, except maybe on the w/e's, when they still close in the afternoon. City Market Coffee House and Scooter's just south of the City Market are pretty decent by all accounts. Scooter's has WiFi, not sure about City Market. Apparently there's not quite enough people living/shopping downtown to have the extended hours students need. The Kansas City Power and Light District hold promise, but right now it's pretty much just bars and such.
The best places I know for sure to study are Broadway Cafe in Westport if you want local flavor, and Starbucks in Barnes and Noble on the Plaza has great ambiance. It's the Barnes and Noble that gives it the charm of course (it's the Plaza!) - my study group and I used to go there a lot for a change in venue.
There were others of course, both in the City Market and Westport, but they've gone the way so many other coffee houses have gone with the proliferation of "a Starbucks on every corner." There may be others downtown, but you'll just have to scope them out yourself.
Part of the issue of why there are so many in the Plaza & Westport area is because all the universities are in that area: KU, UMKC, Rockhurst, UMKC SOM & Dental school (midtown but still closer to Westport than downtown) and KCUMB is the only one downtown. There are plenty of newer ones in those areas I haven't visited; once you get into clerkships/residency, coffee house studying mostly goes by the wayside - you pretty much have to study "on the run."
Anyway maybe those will give you a place to start, and you can explore outward from there! Cherish these next 2 years and make the most of them. Believe it or not, they have the potential to be the most fun (yes, fun!) and exciting time as you travel down the path to becoming a physician. The close friends you make during this phase will likely be friends for life, so cherish them as well!
Good luck as you start this next phase of your exciting quest for knowledge!
Cheers,