14. B.

Let the side length of the regular octagon be $a$. Then
$$
\begin{array}{l}
\frac{m}{n}=\frac{a^{2}+a^{2}-2 a^{2} \cos 135^{\circ}}{a^{2}+a^{2}-2 a^{2} \cos 135^{\circ}+4 \times \frac{1}{2} \times \frac{\sqrt{2}}{2} a^{2}} \\
=\frac{2+\sqrt{2}}{2+\sqrt{2}+\sqrt{2}}=\frac{\sqrt{2}}{2} .
\end{array}
$$
14. Let the area of the regular octagon $A B C D E F G H$ be $n$, and the area of quadrilateral $A C E G$ be $m$. Then the value of $\frac{m}{n}$ is ( ).
(A) $\frac{\sqrt{2}}{4}$
(B) $\frac{\sqrt{2}}{2}$
(C) $\frac{3}{4}$
(D) $\frac{3 \sqrt{2}}{5}$
(E) $\frac{2 \sqrt{2}}{3}$