In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1.$ What is $FI^2$? $\textbf{(A) } \frac{7}{3} \qquad$ $ \textbf{(B) } 8-4\sqrt2 \qquad$ $ \textbf{(C) } 1+\sqrt2 \qquad$ $ \textbf{(D) } \frac{7}{4}\sqrt2 \qquad$ $ \textbf{(E) } 2\sqrt2$