Perhaps "K" wants both you and his girlfriend. This would be great if his girlfriend was informed and consenting. However he is sure that he can't get her consent, so he chooses not to inform her. This will blow up eventually; no affair can be carried on in secret forever. If he's sure that his girlfriend won't consent, then it's his responsibility to make a choice between you and his girlfriend.
You are at least willing to broach the topic with your husband. This gives your husband the chance to decide whether to be consenting. If it would help, you could point out to him that no two men are alike, and that therefore you do get something different things from both "K" and your husband. Sharing poly reading materials with him will probably also help. Franklin Veaux's poly pages
might be a place to start.
But even if your husband (eventually) says "okay," you still have a decision to make. Do you continue to see "K," thus making yourself an accomplice to the secret he is keeping indefinitely from his girlfriend? Do you want to be the secret partner forever? How long should the secret be kept?
I realize these aren't easy questions. Give it some time to think about it ... and as GalaGirl said, be careful.