If you really want to take the "L" train, take the Orange Line train from Midway to the Clark/Lake "L" station in downtown Chicago. Then, to avoid paying a second fare, follow the signs to the Blue Line, which is in the subway below the Orange Line. Take the Blue Line to O'Hare Airport. It will cost $2.50, and take about 90 minutes. The only surcharge that you would experience is if you return, you will have to pay a total of $5 when you leave O'Hare. (There is no surcharge for rides leaving Midway.)
There are also taxis and express buses that connect the airport, but the costs and travel times vary. You can inquire about those connections at the Midway luggage pickup area.