  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 fb$^{-1}$ of integrated luminosity
collected at a center of mass energy of $\sqrt{s}$ = 1.96 TeV with the D0
detector at the Fermilab Tevatron $p\bar{p}$ 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, $WH\to\ell\nu b\bar{b}$; gluon fusion with the Higgs decaying to $W$
boson pairs, $H\to WW\to\ell\nu jj$; and associated production with a vector
boson where the Higgs decays to $W$ boson pairs, $VH\to VWW \to \ell\nu jjjj$
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 $Z\to b\bar{b}$ 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
\obslimA$\times\sigma_{\rm SM}$, where $\sigma_{\rm SM}$ is the standard model
Higgs boson production cross section, while the expected limit is
\explimA$\times\sigma_{\rm SM}$. We also interpret the data considering models
with fourth generation fermions, or a fermiophobic Higgs boson.
