BALKAN 2014 Problem 3

Let $A B C D$ be a trapezium inscribed in a circle $\Gamma$ with diameter $A B$. Let $E$ be the intersection point of the diagonals $A C$ and $B D$. The circle with center $B$ and radius $B E$ meets $\Gamma$ at the points $K$ and $L$, where $K$ is on the same side of $A B$ as $C$. The line perpendicular to $B D$ at $E$ intersects $C D$ at $M$.

Prove that $K M$ is perpendicular to $D L$.