Hi Harman - you could probably walk it in 30 mins. Walk Victoria Street up past the side of Westminster Abbey to Parliament Square. Cross the bridge where Big Ben is - and you'll have a fab view of the London Eye (the Aquarium is in one of the riverfront buildings by the Eye). Once over the bridge, turn right, and the Hospital is along there. (I've never actually been into the hospital, so have no idea where the main entrance is). But not sure you'd want to do that much of a walk.
Alternatively, if you go out the front/main entrance of Victoria station, there are lots of bus lanes. Walk over to the London Transport booth (little grey prefab hut that's off to the right/farthest lane if your back is to the station). There are people there who will tell you what bus to get - I'm sure there must be one direct. They'll give you a bus map too if you want one. Or, if you wanted to take a taxi, would definitely be less than £10.
Would love to meet for a coffee, but we'll be in the midst of our move. Argh! Anyway, hope everything goes well. Good luck with the appointments.