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 $AEH$ has area $\frac{1}{2},$ quadrilateral $BFIE$ has area $1,$ quadrilateral $DHJG$ has area $1,$ and pentagon $FCGJI$ has area $1\frac{1}{2}.$ What is $F{I}^2$?