The area of the circle increases by a factor of 4.

If the lengths all increase by a factor of $k$, then the perimeter will increase by a factor of $k$ and the area by a factor of $k^2$.

If the lengths all increase by a factor of $k$, then the surface area will increase by a factor of $k^2$ and the volume by a factor of $k^3$.

The volume of Jimmy's balloon doubles, so the lengths have all increased by a factor of $\sqrt[3]{2}$. So the surface area has increased by a factor of $\sqrt[3]{4}$, so Jimmy needs $\sqrt[3]{4}L$.