  Twenty years ago, the mismatch between the observed number of Milky Way
satellite galaxies and the predicted number of cold dark matter (CDM) subhalos
was dubbed the ``missing satellites problem". Although mostly framed since in
terms of satellite counts in luminosity space, the missing satellites problem
was originally posed in velocity space. Importantly, the stellar velocity
dispersion function encodes information about the density profile of satellites
as well as their abundance. In this work, we completeness correct the MW
satellite stellar velocity dispersion function down to its ultrafaint dwarfs
($L \gtrsim 340$ L$_\odot$). Our most conservative completeness correction is
in good agreement with a simple CDM model in which massive, classical
satellites (M$_{\rm 200} \gtrsim 5 \times 10^8~$M$_\odot$) have baryon-driven
cores, while lower mass, ultrafaint satellites inhabit cuspy halos that are not
strongly tidally stripped. Tidal destruction of satellites by the MW's disk
must be minimal, otherwise the completeness-corrected velocity function exceeds
any plausible CDM prediction -- a ``too many satellites" problem. We rule out
non-core-collapsing self-interacting dark matter models with a constant cross
section $\gtrsim$ 0.1 cm$^2$/g. Constraints on warm dark matter are stronger
than those based on the luminosity function due to its additional sensitivity
to subhalo central densities, which suppresses number counts by up to an
additional 30%. A thermal relic mass $\gtrsim$ 6 keV is preferred. Reducing
uncertainties on stellar velocity dispersion measurements and the amount of
tidal stripping experienced by the faintest dwarfs is key to determining the
severity of the too many satellites problem.
