  We consider the axial-vector mesons $K_1(1270)$ and $K_1(1400)$ as a mixture
of two $|^3P_1\rangle$ and $|^1P_1\rangle$ states with the mixing angle
$\theta$ that equal to $(-34\pm 13)^\circ$. We calculate the light-front
distribution amplitudes (LFDAs) and decay constant formulas for both the
axial-vector mesons $K_1$ in the AdS/QCD correspondence. The transition form
factors of the semileptonic $B\to K_1$ decays are derived in terms of the LFDAs
for $K_1$ mesons. Using these form factors and decay constant values, the
differential branching ratios of $B\to K_1 (1270, 1400) \ell^+
\ell^-$,~$\ell=\mu,\tau$ transitions are plotted with respect to the
four-momentum transfer squared, $q^2$. In addition, the branching ratio values
of these decays and the non-leptonic $B \to K_1(1270, 1400) \gamma$ decays are
estimated. A comparison is made between our results for the branching ratios of
$B\to K_1(1270, 1400) \gamma$ decays in the AdS/QCD model and predictions
obtained from the light-cone sum rules (LCSR) as well as the experimental
values. Finally, the forward-backward asymmetries for the aforementioned
semileptonic decays are plotted on $q^2$ in both the AdS/QCD correspondence and
two Higgs doublet model (2HDM) in order to test the standard model (SM) and
search for the new physics (NP).
