I know how to do the first part, i think... (finding \sigma). Basically you just use Gauss's Law
\oint{E \cdot dA}=\frac{Q_\text{enclosed}}{\epsilon_0}
... except you set the r in that equation equal to 2R, so you can ultimately set the expression for the E-field equal to zero and...
A solid sphere of radius R has a non-uniform volume charge density \rho(r) and a constant surface charge density \sigma. If the field inside the sphere is uniform and radially atuned, and the field a distance 2R away from the center is zero, find \rho and \sigma in terms of R, r (distance from...