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 2, quadrilateral $BFIE$ has area 2, quadrilateral $DHJG$ has area 2, and pentagon $FCGJI$ has area 10. What is $F{I}^2$?