A search is presented for the production of vector-like quark pairs, T$\overline{\mathrm{T}}$ or Y$\overline{\mathrm{Y}}$, with electric charge of 2/3 (T) or -4/3 (Y), in proton-proton collisions at $\sqrt{s} =$ 13 TeV. The data were collected by the CMS experiment at the LHC in 2016 and correspond to an integrated luminosity of 35.8 fb$^{-1}$. The T and Y quarks are assumed to decay exclusively to a W boson and a b quark. The search is based on events with a single isolated electron or muon, large missing transverse momentum, and at least four jets with large transverse momenta. In the search, a kinematic reconstruction of the final state observables is performed, which would permit a signal to be detected as a narrow mass peak ($\approx$7% resolution). The observed number of events is consistent with the standard model prediction. Assuming strong pair production of the vector-like quarks and a 100% branching fraction to bW, a lower limit of 1295 GeV at 95% confidence level is set on the T and Y quark masses.