We present a search for the standard model Higgs boson in final states with a charged lepton (electron or muon), missing transverse energy, and two or three jets, at least one of which is identified as a $b$-quark jet. The search is primarily sensitive to $WH\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}b\overline{b}$ production and uses data corresponding to $9.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity collected with the D0 detector at the Fermilab Tevatron $p\overline{p}$ Collider at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$. We observe agreement between the data and the expected background. 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 $5.2\ifmmode\times\else\texttimes\fi{}{\ensuremath{\sigma}}_{\mathrm{SM}}$, where ${\ensuremath{\sigma}}_{\mathrm{SM}}$ is the standard model Higgs boson production cross section, while the expected limit is $4.7\ifmmode\times\else\texttimes\fi{}{\ensuremath{\sigma}}_{\mathrm{SM}}$.