Note that $AC\leq BD\iff \angle A\geq 90$. Clearly any point $A$ in the circle works. Now if $A$ is outside, the maximum possible $\angle A$ is when $AB,AD$ are tangents to the circle. So $A$ must be in the circle with the same center as $K$ but the radius is $\sqrt2$ times that of $K$, and the locus is all the points in that circle.