  We construct the metric perturbation in Lorenz gauge for a compact body on a
circular equatorial orbit of a rotating black hole (Kerr) spacetime, using a
newly-developed method of separation of variables. The metric perturbation is
formed from a linear sum of differential operators acting on Teukolsky mode
functions, and certain auxiliary scalars, which are solutions to ordinary
differential equations in the frequency domain. For radiative modes, the
solution is uniquely determined by the $s=\pm2$ Weyl scalars, the $s=0$ trace,
and $s=0,1$ gauge scalars whose amplitudes are determined by imposing
continuity conditions on the metric perturbation at the orbital radius. The
static (zero-frequency) part of the metric perturbation, which is handled
separately, also includes mass and angular momentum completion pieces. The
metric perturbation is validated against the independent results of a 2+1D time
domain code, and we demonstrate agreement at the expected level in all
components, and the absence of gauge discontinuities. In principle, the new
method can be used to determine the Lorenz-gauge metric perturbation at a
sufficiently high precision to enable accurate second-order self-force
calculations on Kerr spacetime in future. We conclude with a discussion of
extensions of the method to eccentric and non-equatorial orbits.
