jmc

Quadrilateral $KLMN$ is a square because it has $90^{\circ }$ rotational symmetry, which implies that each pair of adjacent sides is congruent and perpendicular. Since $ABCD$ has sides of length 4 and $K$ is $2\sqrt{3}$ from side $\overline{AB}$, the length of the diagonal $\overline{KM}$ is $4+4\sqrt{3}$. Since the area of a square is half the product of its diagonals, the area is $$\frac{1}{2}(4+4\sqrt{3}{)}^2=\boxed{32+16\sqrt{3}}.$$