My rook was targeting the opponent's king but her knight was in between
This is what is called an (absolute) pin in chess, i.e. the knight is pinned by the rook and cannot move, because it would put the king in check (which is not allowed).
I moved my knight to the check place checked by same knight
The way I understand it, what you mean is something like: I moved my knight to a square attacked by the pinned knight. Of course this is perfectly legal as there are no restrictions put on the movement of your knight.
Can I move my king to that place
No, you cannot move your king to a square attacked by an enemy piece, even if the piece is pinned.
If you think of "winning in chess" as "who captures the enemy king first", all of this makes sense. Specifically, the pinned knight would move first to capture your king, just before you could capture her king (which is now under attack since the knight has moved).
If you search a bit on stackexchange you will find many questions regarding this problem, such as: this question.