Radius of a circle

Let $ABCD$ be a line with the four points $A, B, C, D$ in that order such that $AB=BC=CD=4$ . Let $E, F, G, H$ be points such that $ABFE, BCGF, CDHG$ are rectangles. Then let $J, K$ be points such that $GFJK$ is a rectangle and $FJ=FB=x$ .

If $x$ is a positive integer, and the radius of the circle passing through $A, D, J, K$ is the square root of a non-square positive integer, find $r+S$ where $\sqrt {r}$ is the largest possible radius of the circle and $S$ is the sum of all possible values of $x$ .