  Vacuum perturbations of the Kerr metric can be reconstructed from the
corresponding perturbation in either of the two Weyl scalars $\psi_0$ or
$\psi_4$, using a procedure described by Chrzanowski and others in the 1970s.
More recent work, motivated within the context of self-force physics, extends
the procedure to metric perturbations sourced by a particle in a bound geodesic
orbit. However, the existing procedure leaves undetermined a certain
stationary, axially-symmetric piece of the metric perturbation. In the vacuum
region away from the particle, this "completion" piece corresponds simply to
mass and angular-momentum perturbations of the Kerr background, with amplitudes
that are, however, a priori unknown. Here we present and implement a rigorous
method for finding the completion piece. The key idea is to impose continuity,
off the particle, of certain gauge-invariant fields constructed from the full
(completed) perturbation, in order to determine the unknown amplitude
parameters of the completion piece. We implement this method in full for bound
(eccentric) geodesic orbits in the equatorial plane of the Kerr black hole. Our
results provide a rigorous underpinning of recent results by Friedman {\it et
al.}\ for circular orbits, and extend them to non-circular orbits.
