It is about 23 miles on the road (mostly Cicero Ave).
The taxi will take anywhere from 38 minutes to 1:15 depending on traffic. It will cost you at least 4 times as much as using the "El".
The El route involves transferring from the orange line to the blue line in downtown Chicago. There is no worry of missing your stop since Midway is at the end of the orange and O'Hare is the end of the blue. The train route would take at least 1:30 including the transfer.

Hi, goodvibrations.

I think you'd be better off biting the bullet and paying for a cab/car. I recall one journey from Midway (on the orange line) to the Loop/downtown took me about 45 minutes, and sometimes the train from the Loop to O'Hare (on the blue line can run anywhere from 35 minutes to 50-55), then you need to add about 10 minutes to get your bearings after getting from the elevated tracks (orange line) down to the subway (blue line) at the Clark/Lake stop.

And tomfuller is right about the blue line ending at O'Hare, and it's unlikely you'll miss the Clark/Lake stop on the Orange line, but the elevated orange line does "loop" around the downtown area. It doesn't come to a dead-end stop.

My company uses A-1 Car service (they're reliable), but Carey Cars is around, too. Maybe give one of them a call and ask their travel time estimate.

http://a1limousine.com/

Oh! And then the some international flights leave from the "main"/domestic terminals and others leave from the detached international terminal that you take the tram to. THAT adds time.

Sorry... I'm such a bummer.

Best of luck!

Oh geesh! I see you didn't want to take a train anyway. So sorry to junk up the posting.

Hopefully someone can speak more knowledgeably than I on car/taxi service between the two. My apologies.