We present, in detail, a search for the standard model Higgs boson, $H$, in final states with a charged lepton (electron or muon), missing energy, and two or more jets in data corresponding to $9.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity collected at a center-of-mass energy of $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ with the D0 detector at the Fermilab Tevatron $p\overline{p}$ Collider. The search uses $b$-jet identification to categorize events for improved signal versus background separation and is sensitive to associated production of the $H$ with a $W$ boson, $WH\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}b\overline{b}$; gluon fusion with the Higgs decaying to $W$-boson pairs, $H\ensuremath{\rightarrow}WW\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}jj$; and associated production with a vector boson where the Higgs decays to $W$-boson pairs, $VH\ensuremath{\rightarrow}VWW\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}jjjj$ production (where $V=W$ or $Z$). We observe good agreement between data and expected background. We test our method by measuring $WZ$ and $ZZ$ production with $Z\ensuremath{\rightarrow}b\overline{b}$ and find production rates consistent with the standard model prediction. For a Higgs boson mass of 125 GeV, we set a 95% C.L. upper limit on the production of a standard model Higgs boson of $5.8\ifmmode\times\else\texttimes\fi{}{\ensuremath{\sigma}}_{\mathrm{SM}}$, where ${\ensuremath{\sigma}}_{\mathrm{SM}}$ is the standard model Higgs boson production cross section, while the expected limit is $4.7\ifmmode\times\else\texttimes\fi{}{\ensuremath{\sigma}}_{\mathrm{SM}}$. We also interpret the data considering models with fourth generation fermions, or a fermiophobic Higgs boson.