Interesting, so it seems like your circuit is actually radial not ring, or they are both spurs. Or they are spurs off a 4mm radial.
It looks like someone has left a loop of cable past the plinth socket, for some unknown reasons. Perhaps that is the ring, perhaps not.
One simple option would be to daisy chain the new socket to the washing machine socket. This would limit it from being used from anything powerful while the washing machine is on, but if it's for lights it should be ok.
Look in your consumer unit and take a photo. Work out by switching one at a time which circuit breaker powers the kitchen socket. See if it says C20 or C32.
If C20, you can connect to the double socket or Fcu input side. If C32 then not, you can only connect the fcu output side.