jmc

If we let the side of length $2s$ be the base of the parallelogram, we can use our 45-degree angle to find the height of the parallelogram. The height and sides of the parallelogram form a 45-45-90 triangle, with the side of length $s$ as the hypotenuse. Thus, the height of the parallelogram is $s/\sqrt{2}$.

It follows that the area of the parallelogram is $2s\cdot (s/\sqrt{2})=s^2\sqrt{2}=8\sqrt{2}$, so $s^2=8$. Taking the square root of both sides, we see that $s=\boxed{2\sqrt{2}}$.