There are a few things wrong with your formulas:
Your total energy integral should look like this:
E = 2 \int_0^{k_f} \frac{\hbar^2 k^2}{2m} \frac{V}{(2\pi)^3} 4 \pi k^2 dk
The \frac{V}{(2\pi)^3} is necessary because you need to account for the volume of...