I would say it is about 15 miles from the Contemporary. If the taxi goes out Main Gate and East on 192 you will also pass through a lot of construction. Less time but higher mileage (plus tolls) would be taking the Osceola Parkway most of the way.
Taxi fares currently are $2.00 at the flag drop (1/4 mile) and then $0.25 per 1/8 mile. Including tip, you would be looking at about $35 each way.