  A search for dark matter particles is performed using events with a pair of W
bosons and large missing transverse momentum. Candidate events are selected by
requiring one or two leptons ($\ell =$ electrons or muons). The analysis is
based on proton-proton collision data collected at a center-of-mass energy of
13 TeV by the CMS experiment at the LHC and corresponding to an integrated
luminosity of 138 fb$^{-1}$. No significant excess over the expected standard
model background is observed in the $\ell\nu$qq and 2$\ell$2$\nu$ final states
of the W$^+$W$^-$ boson pair. Limits are set on dark matter production in the
context of a simplified dark Higgs model, with a dark Higgs boson mass above
the W$^+$W$^-$ mass threshold. The dark matter phase space is probed in the
mass range 100-300 GeV, extending the scope of previous searches. Current
exclusion limits are improved in the range of dark Higgs masses from 160 to 250
GeV, for a dark matter mass of 200 GeV.
