The particle is moving in a circle. Think about the gravitational potential in a circular orbit centered on a mass, is it not the same everywhere? And if the potential is constant, with reference to what? You can safely set V = 0.
Energy:
Use your Hamiltonian (derive twice, set H\psi = E\psi...
Angular momentum:
I believe you should have shown in b) that the stationary wavefunction is a eigenfunction for the angular momentum operator. Your eigenvalue should contain your quantization number k.
Energy:
Use the hamiltonian operator on the stationary wavefunction, and you should find...
This is quite similar to the particle in a box problem. Also, your last integral is incorrect as you suspected, N_k^2\int_0^{2\pi}d\phi = 2\pi N_k^2. Finding N_k should not be very hard :-)