  The upcoming LISA mission offers the unique opportunity to study the Milky
Way through gravitational wave radiation from Galactic binaries. Among the
variety of Galactic gravitational wave sources, LISA is expected to
individually resolve signals from $\sim 10^5$ ultra-compact double white dwarf
(DWD) binaries. DWDs detected by LISA will be distributed across the Galaxy,
including regions that are hardly accessible to electromagnetic observations
such as the inner part of the Galactic disc, the bulge and beyond. We
quantitatively show that the large number of DWD detections will allow us to
use these systems as tracers of the Milky Way potential. We demonstrate that
density profiles of DWDs detected by LISA may provide constraints on the scale
length parameters of the baryonic components that are both accurate and
precise, with statistical errors of a few percent to $10$ percent level.
Furthermore, the LISA sample is found to be sufficient to disentangle between
different (commonly used) disc profiles, by well covering the disc out to
sufficiently large radii. Finally, up to $\sim 80$ DWDs can be detected through
both electromagnetic and gravitational wave radiation. This enables
multi-messenger astronomy with DWD binaries and allows one to extract their
physical properties using both probes. We show that fitting the Galactic
rotation curve constructed using distances inferred from gravitational waves
{\it and} proper motions from optical observations yield a unique and
competitive estimate of the bulge mass. Instead robust results for the stellar
disc mass are contingent upon knowledge of the Dark Matter content.
