Can a knight become a king?

It depends on what exactly you mean by that, but probably "no."

The female equivalent of "knight" is usually "dame" or "lady".

Also, for the most part such an honor would never be granted to a "girl" (or boy), but rather to an adult. Even HRH The Princess Royal (Anne, daughter of Queen Elizabeth) wasn't admitted to a Chivalric Order until she was 21. (Prince Charles was made a Royal Knight of the Order of the Garter when he was 10 or so, but that's a special case: that's when he became Prince of Wales, and the Prince of Wales is automatically a member of the Order of the Garter).

Countries other than the UK can of course have their own rules which may differ widely.