The airlines that fly from JFK to Midway in Chicago are Delta and Sun Country. Neither offer nonstop flights for this route at this time. Nonstop flights are available for JFK to Chicago ORD on American, Delta, and JetBlue.
There are several airlines that make the trip from Dallas to Chicago Midway, however none are non-stop at this time. They include Delta, Frontier, and AirTran airways.
Try Southwest Airlines
Southwest should be one of the carriers.
Unfortunatly I don't know.
US Airways offers nonstop air service from Chicago O'Hare to Sarasota/Bradenton airport.
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Midway Airport
Chicago Midway: IATA: MDW ICAO: KMDW
Midway Airport is approximately 10 miles from downtown Chicago.
Midway Airport is in Chicago; it's about 11 miles from the heart of downtown.
Midway International Airport is located in Chicago, Illinois. It is the second largest airport in the entire state of Illinois, after Chicago O'Hare International Airport, and is the largest airport in metropolitan Chicago.
Great Lakes is approximately 35 miles from Chicago Midway Airport.
The Monsters of the Midway. The "Monsters of the Midway".
Chicago Midway International Airport.