Maybe a cheap as chips hire car might be an option. I know the GE is a pricey ride.
Don't use the Gatwick Express. a) its expensive and b) it won't stop until it gets to Victoria. And even if you do want to get to Victoria a normal network southeast train stops at the airport every 20mins or so.
As termie has said, get to clapham and change, I think worcester park may be the nearest station to the Hospital and get a taxi.
If you did want to hire a car, it will take between 40 mins and an hour to drive there from the airport depending on traffic.
