KöMaL problem A, 2019

In quadrilateral $ABCD$, $AB=BC=\frac1{\sqrt2}DA$, and $\angle ABC$ is a right angle. The midpoint of side $BC$ is $E$, the orthogonal projection of $C$ on $AD$ is $F$, and the orthogonal projection of $B$ on $CD$ is $G$. The second intersection point of circle $BCF$ (with center $H$) and line $BG$ is $K$, and the second intersection point of circle $BHC$ and line $HK$ is $L$. The intersection of lines $BL$ and $CF$ is $M$. The center of the Feurbach circle of triangle $BFM$ is $N$. Prove that $\angle BNE$ is a right angle.