  The precision study of $W^-W^+H$ production with subsequent $W^{\pm}
\rightarrow l^{\pm} \overset{ _{(-)}}{\nu_{l}}$ and $H \rightarrow b\bar{b}$
decays at the LHC can help us to study the Higgs gauge couplings and to search
for new physics beyond the SM. In this paper, we calculate the shower-matched
NLO QCD correction and the EW corrections from the $q\bar{q}$ annihilation and
photon-induced channels to the $W^-W^+H$ production at the $14~ {\rm TeV}$ LHC,
and deal with the subsequent decays of Higgs and $W^{\pm}$ bosons by adopting
the {\sc MadSpin} method. Both the integrated cross section and some kinematic
distributions of $W^{\pm}$, $H$ and their decay products are provided. We find
that the QCD correction enhances the LO differential cross section
significantly, while the EW correction from the $q\bar{q}$ annihilation channel
obviously suppresses the LO differential cross section, especially in the high
energy phase-space region due to the Sudakov effect. The $q\gamma$- and
$\gamma\gamma$-induced relative corrections are positive, and insensitive to
the transverse momenta of $W^{\pm}$, $H$ and their decay products. These
photon-induced corrections compensate the negative $q\bar{q}$-initiated EW
correction, and become the dominant EW contribution as the increment of the
$pp$ colliding energy. The parton shower (PS) effects on the kinematic
distributions are nonnegligible. The PS relative correction to the $b$-jet
transverse momentum distribution can exceed $100\%$ in the high $p_{T, b}$
region. We also investigate the scale and PDF uncertainties, and find that the
theoretical error of the ${\rm QCD}+{\rm EW}+q\gamma+\gamma\gamma$ corrected
integrated cross section mainly comes from the renormalization scale dependence
of the QCD correction.
