a few questions - thinking of this as if it were checkers on a checker board, can they jump diagonal? you said L in the center of K start and K end, so (back to the checker board) if there is a space between K and L, can K jump L and leave a space on the other side? i.e. K-L-- becomes --L-K must there only be one jump at a time? or can this happen -KL- becomes L--K (of course this blurs which piece is K and which is L, because they are both K and both L).
ignoring those questions and assuming that one can only jump an adjacent checker and that one can jump diagonally, start by considering your end result.
as you see, it is quite impossible to jump from this back to
because as you can see in the first diagram, (3|0) and (2|-1) may only jump each other diagonally and the same for (0|0) and (1|1), and further, since they diagonals are not adjacent, each set of pieces will always only be in there separate diagonals, therefore we can not possibly arrive at the starting configuration.