jmc

Since the cube has six sides, each of area $2^2=4$, the surface area of the cube is 24. Since Joe's paint will exactly cover the cube and it will also exactly cover the sphere, the sphere also must have surface area 24.

If $r$ is the radius of the sphere, this tells us that $$4\pi r^2=24,$$or $r^2=6/\pi$, so $$r=\sqrt{6/\pi }=\sqrt{6}/\sqrt{\pi }.$$Therefore the volume of the sphere is $$\frac{4}{3}\pi r^3=\frac{4}{3}\pi (\frac{\sqrt{6}}{\sqrt{\pi }}{)}^3=\frac{4}{3}\pi \cdot \frac{6\sqrt{6}}{\pi \sqrt{\pi }}=\frac{8\sqrt{6}}{\sqrt{\pi }}.$$Thus $\boxed{K=8}$.