smc

Let $R$ be the radius of the circle. Draw the two radii that meet the points of tangency to the lines $y=\pm x$. We can see that a square is formed by the origin, two tangency points, and the center of the circle. The side lengths of this square are $R$ and the diagonal is $k=R+6$. The diagonal of a square is $\sqrt{2}$ times the side length. Therefore, $R+6=R\sqrt{2}\Rightarrow R=\frac{6}{\sqrt{2}-1}=6+6\sqrt{2}\Rightarrow \boxed{6+6\sqrt{2}}$.