For a uniform sphere, k = 2/5 = 0.4. This is a bit counterintuitive, since we'd expect a fundamental shape like a sphere to have some nice neat value like 1 or 1/2, but there it is.
The earth's moment of inertia enters into formulas for precession, so we can determine the value of k for the real earth. Since the earth is denser near the center, k is smaller than 0.4. It's 0.33, or a hair less than 1/3.
For a thin spherical shell, k = 2/3
For a flat disk, k=1/2