  High precision proper motion (PM) measurements are available for
approximately 20% of all known dwarf satellite galaxies of the Milky Way (MW).
Here we extend the Bayesian framework of Patel et al. (2017b) to include all MW
satellites with measured 6D phase space information and apply it with the
Illustris-Dark simulation to constrain the MW's mass. Using the properties of
each MW satellite individually, we find that the scatter among mass estimates
is reduced when the magnitude of specific orbital angular momentum (j) is
adopted rather than their combined instantaneous positions and velocities. We
also find that high j satellites (i.e. Leo II) constrain the upper limits for
the MW's mass and low j satellites rather than the highest speed satellites
(i.e. Leo I and LMC), set the lower mass limits. When j of all classical
satellites is used to simultaneously estimate the MW's mass, we conclude the
halo mass is $0.85^{+0.23}_{-0.26}\times 10^{12} \rm \, M_{\odot}$ (including
Sagittarius dSph) and $0.96^{+0.29}_{-0.28} \times 10^{12} \rm \, M_{\odot}$
(excluding Sagittarius dSph), cautioning that low j satellites on decaying
orbits like Sagittarius dSph may bias the distribution. These estimates
markedly reduce the current factor of two spread in the mass range of the MW.
We also find a well-defined relationship between host halo mass and satellite j
distribution, which yields the prediction that upcoming PMs for ultra-faint
dwarfs should reveal j within $ 5\times10^3 - 10^4$ kpc km s$^{-1}$. This is a
promising method to significantly constrain the cosmologically expected mass
range for the MW and eventually M31 as more satellite PMs become available.
