The only reason for the G's to have recruited Kate is to nullify her ability to kill them on instructions from the detective. Because Kate and Ford came out with their roles, they would have to either whack her, or recruit her, but she was jailed so whacking was out of the question. Now Ford isn't recruitable, which means the G's could put Kate, Ford, and a third confirmed C (Voller or ILFJ) as their pool for recruiting. Which means they either got Kate, or one of ILFJ or Voller.
If Kate is now a G, she cannot kill someone. If she is not a G, one of ILFJ and Voller is the recruited G. She can kill one of those two to prove it was not her recruited, if she can't do so, she is a G, and we lynch her the next day. The problem is still deciding which one of ILFJ and Voller would have been put in the pool with Kate? On this, I am stumped. If Kate kills a G, she gets a third shot, and at some point one can only hope Ford will investigate a G, instead of a C.