Hail Stewart-2!

Geometry Level 2

Let $ABCD$ be a square, and let $E$ and $F$ be points on $AB$ and $BC,$ respectively. The line through $E$ parallel to $BC$ and the line through $F$ parallel to $AB$ divide $ABCD$ into two squares and two non-square rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD.$

Find $\frac{AE}{EB}+\frac{EB}{AE}.$