... I suspect this is going a bit too deep, but it occurred to me that if it's been established that the killer was either King or Mimi, one can understand King trying to frame Mimi - but why should she need to frame him? She knows that she didn't do it, so she must have guessed that he did and could perhaps have assumed that the truth will out.

Yes, it is a bit too deep, isn't it, though I personally did share King's dislike of Mimi, even though we know a lot less about her than about him.