Higgs boson production via gluon-gluon fusion and vector-boson fusion in proton-proton collisions is measured in the $H\ensuremath{\rightarrow}W{W}^{*}\ensuremath{\rightarrow}e\ensuremath{\nu}\ensuremath{\mu}\ensuremath{\nu}$ decay channel. The Large Hadron Collider delivered proton-proton collisions at a center-of-mass energy of 13 TeV between 2015 and 2018, which were recorded by the ATLAS detector, corresponding to an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The total cross sections for Higgs boson production by gluon-gluon fusion and vector-boson fusion times the $H\ensuremath{\rightarrow}W{W}^{*}$ branching ratio are measured to be $12.0\ifmmode\pm\else\textpm\fi{}1.4$ and $0.75{\text{ }}_{\ensuremath{-}0.16}^{+0.19}\text{ }\text{ }\mathrm{pb}$, respectively, in agreement with the Standard Model predictions of $10.4\ifmmode\pm\else\textpm\fi{}0.6$ and $0.81\ifmmode\pm\else\textpm\fi{}0.02\text{ }\text{ }\mathrm{pb}$. Higgs boson production is further characterized through measurements of Simplified Template Cross Sections in a total of 11 kinematic fiducial regions.