Begin by noticing that $K$ is convex: Otherwise, the convex hull $\hat{K}$ of $K$ would have a side $ab$ whose line of support is a diagonal line of $K$ ($a$ and $b$ are, of course, non-adjacent vertices of $K$ and there might virtually be other vertices or even sides of $K$ along the line segment $ab$). The reflexion of a third vertex of $\hat{K}$, and hence of $K$, in the line $ab$ would then fall outside $\hat{K}$, and hence outside $K$, contradicting the vertex reflexion condition $K$ satisfies.

Next, let $a$, $b$ and $c$ be consecutive vertices of $K$, and let $d \ne a, b, c$ be a fourth vertex. Since the reflexions of $a$ and $c$ in the line $bd$ do not fall outside $K$, it follows that $bd$ is the internal bisectrix of the angle $abc$, and, by convexity, $a$ and $c$ are reflexions of one another in the line $bd$.

Consequently, $K$ is a convex quadrangle $abcd$ such that $a$ and $c$ are reflexions of one another in the line $bd$, and $b$ and $d$ are reflexions of one another in the line $ac$; that is, $K$ is a lozenge (rhombus).