This Letter describes the search for an enhanced production rate of events with a charged lepton and a neutrino in high-energy $pp$ collisions at the LHC. The analysis uses data collected with the CMS detector, with an integrated luminosity of $5.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$, and a further $3.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$. No evidence is found for an excess. The results are interpreted in terms of limits on a heavy charged gauge boson (${W}^{\ensuremath{'}}$) in the sequential standard model, a split universal extra dimension model, and contact interactions in the helicity-nonconserving model. For the last, values of the binding energy below 10.5 (8.8) TeV in the electron (muon) channel are excluded at a 95% confidence level. Interpreting the $\ensuremath{\ell}\ensuremath{\nu}$ final state in terms of a heavy ${W}^{\ensuremath{'}}$ with standard model couplings, masses below 2.90 TeV are excluded.