An algorithm is developed for performing calculations for the nonlocal electrostatic solvation theory of an ion in a cavity, accounting for electrostatic boundary conditions. The latter implies an induced charge distribution on the cavity surface as well as an induced volume charge distribution in the medium. This approach is validated by a variational derivation which also provides a general expression for the solvation energy. The procedure, implemented for spherical ions, is tested by calculating the analytic solution for an exponential nonlocal dielectric kernel and determining the corresponding solvation energy. Parametrization is presented for a range of solvents, fitted to experimental solvation energies.