AMC 12 2014b Problem 21

In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?

pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2);
draw(A--B--C--D--cycle);
draw(K--H--G--J--cycle);
draw(F--E);
label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N);
label("$G$",G,E); label("$H$",H,N); label("$J$",J,S); label("$K$",K,W);

$\textbf{(A) }\frac{1}{2}(\sqrt{6}-2) \textbf{(B) }\frac{1}{4} \textbf{(C) }2-\sqrt{3} \textbf{(D) }\frac{\sqrt{3}}{6} \textbf{(E) } 1-\frac{\sqrt{2}}{2}$