Which bit of Kennington?
The E&C suggestion is OK but the diversion around it is a bit slow. I live in the area, and the way I would go is:
Mark Lane (2 way for bikes)
CS3 to Blackfriars Bridge
CS6 to Lambeth Road (by the Imperial War Museum)
Lambeth Road (if I didn't have the kids with me) or use the Pedestrian Crossing then cut through the IWM gardens (with kids)
Kenington Road (bus lane)
If I was in tourist mode I would carry along CS3 all the way to the Houses of Parliament, then:
Millbank
Vauxhall Bridge (segregated cycle lane)
A3024 / Kennington Lane (or Vauxhall Pleasure Gardens, Tyers St, Sancroft St)