Other than using a different card as Chris suggested, there’s no way of splitting the journey without waiting 20 minutes at Vauxhall. You could touch in on a bus, but that would add more than you save.
The other possibility is to wait at Kings Cross St Pancras until 0927 before touching in. Off-peak starts a little early to avoid issues with station clocks being out of sync. The TfL journey planner says you should start at Kings Cross at 0923 to be able to make the 0951 from Vauxhall. If you can walk fast then you should be able to make up the lost time. Trains to Chessington leave from platform 8 at Vauxhall.