Kinda sounds like you might have a couple of simple questions to ask of K: "Are you committed to this relationship?" and, "Will you be staying with us for the long haul?" Even if K doesn't know the answer right away, at least it gives him something to think about and you can always re-iterate the questions later.
Seems to me like you guys are pretty happy with the relationship you already have, other than those insecurities about K's commitment level. So, no need to make things complicated, I guess is what I'm saying.
Love means never having to say, "Put down that meat cleaver!"