  We present the first fully relativistic study of gravitational radiation from
bodies in circular equatorial orbits around the massive black hole at the
Galactic Center, Sgr A* and we assess the detectability of various kinds of
objects by the gravitational wave detector LISA. Our computations are based on
the theory of perturbations of the Kerr spacetime and take into account the
Roche limit induced by tidal forces in the Kerr metric. The signal-to-noise
ratio in the LISA detector, as well as the time spent in LISA band, are
evaluated. We have implemented all the computational tools in an open-source
SageMath package, within the Black Hole Perturbation Toolkit framework. We find
that white dwarfs, neutrons stars, stellar black holes, primordial black holes
of mass larger than $10^{-4} M_\odot$, main-sequence stars of mass lower than
$\sim 2.5\, M_\odot$ and brown dwarfs orbiting Sgr A* are all detectable in one
year of LISA data with a signal-to-noise ratio above 10 during at least $10^5$
years in the slow inspiral towards either the innermost stable circular orbit
(compact objects) or the Roche limit (main-sequence stars and brown dwarfs).
The longest times in-band, of the order of $10^6$ years, are achieved for
primordial black holes of mass $\sim 10^{-3} M_\odot$ down to $10^{-5}
M_\odot$, depending on the spin of Sgr A*, as well as for brown dwarfs, just
followed by white dwarfs and low mass main-sequence stars. The long time
in-band of these objects makes Sgr A* a valuable target for LISA. We also
consider bodies on close circular orbits around the massive black hole in the
nucleus of the nearby galaxy M32 and find that, among them, compact objects and
brown dwarfs stay for $10^3$ to $10^4$ years in LISA band with a 1-year
signal-to-noise ratio above 10.
