Density of an electron gas in 3D using surface integral

I am trying to derivative the state density using the method I mention in my title.

This is how I begin :

$$V/(2pi)^3 cdot int_0^EdS_k/|nabla_kE(k)|.$$

I use spherical coordinates to express the surface element of the sphere with radius k, in the momentum space :

$$dA=k^2sin(theta)dtheta dPhi$$

and also the energy dispersion relation for a free particle (electron) :

$$E=hbar^2k^2/2m$$

In the end i get the following equation for the state density :

$$D(E)=Vcdot sqrt{2}E^2/2pi^2(hbar c)^3 .$$

I think the 2 should not be there.
Maybe I have to multiply with 2 because electrons can be with spin up and spin down in the same state?