Within a certain age bracket, many American girls who prefer to be perceived as cute and girly, simply type the letters "kk" to exude a sense of affable agreeableness. This is often paired with "oo" to present surprise or acquired understanding.

I get the sense that American girls are doing it both to promote a youthful personality, and because a double-tap of the same letter is efficient, reducing effort while typing. It's also used by older women adopting a cutsey voice or tone, especially women raising teenage daughters, sometimes as a mock teeny bopper act.

Males almost never ever use this regardless of age, unless engaging in feminine play as parody, and it gets dropped among females cognizant of their perception as adult women retaining possession of responsibilities and authority.

Probably, among all non-users, "kk" is dropped in fearful avoidance of the dreaded accidental "kkk" in an unnacceptable context.

Among all, a curt use of "k" with a cessation of further conversation is sometimes an expression of annoyance, either because the conversation took a turn they didn't like, or because they're busy or being interrupted somehow.

Capital "K" in this sort of context, combined with habitually minimized capitalization and little or no grammatical punctuation as a personality trait, is almost always an expression of sulking, pouting or open anger with consequences pending. Capital "K" after receiving bad news is as if to say "someone will pay for this, maybe you."