Why not just get the train from Cambridge to Kings Cross? Then tube it around London from there.
This, but forget the Tube.
It is not extravagant to use cabs if you only go to London once or twice a year.
On the rare occasions I go to London, I will train to Waterloo and then walk or Cab, if it costs me £70 over the day then fine. I would rather do that than use the Tube!
Tube is HORRIBLE
(I dare say it is something that people who have to do so will force themselves to get used to, but for a country soul. The tube is intolerable! Cabs every time for me!)