There's also the 59 bus, which goes from King's Cross to Waterloo Bridge, a short walk along the river path from St Thomas's, if you don't fancy the Tube. It's pretty quick, considering.
If you do get the Tube, the easiest way is to get the Victoria Line to Oxford Circus, where it's an easy (30 second) change to the Bakerloo Line. At Waterloo, take the South Bank exit (not the station exit), and then you can just walk along York Road to the hospital.
A final option is to get the Circle Line all the way round to Westminster, which is the same distance from St Thomas's main entrance as Waterloo. It'll take longer, but is easier if you're not feeling too well (and there should be lifts at both stations).
Driving at peak times, well... The A1 is a nightmare at Buckden, Black Cat and Sandy (and to a lesser extent, Biggleswade) in the morning. I'd avoid, personally.