Finding constant in electrostatic equation given only charge
Why electric field is zero? It is a gradient of potential
$$
E=-grad(V)
$$
Now use Maxwell's equation for electric field taking into account that there is now electric charge in your space
$$
\mathrm{div}\,E = 0
$$
Then combine the two equations
$$
\mathrm{div}\,\mathrm{grad}\, V=\Delta V=0
$$
Where \$\Delta\$ is a Laplace operator.
Apply the operator to your equation for V and you should be able to solve it for k.