We consider the axial-vector mesons $$K_1(1270)$$ $K_1\left(1270\right)$ and $$K_1(1400)$$ $K_1\left(1400\right)$ as a mixture of two $$|^3P_1\rangle $$ ${|}^3{P}_1⟩$ and $$|^1P_1\rangle $$ ${|}^1{P}_1⟩$ states with the mixing angle $$\theta $$ $\theta$ that equal to $$(-34\pm 13)^\circ $$ ${(-34\pm 13)}^{\circ }$ . We calculate the light-front distribution amplitudes (LFDAs) and decay constant formulas for both the axial-vector mesons $$K_1$$ $K_1$ in the AdS/QCD correspondence. The transition form factors of the semileptonic $$B\rightarrow K_1$$ $B\to K_1$ decays are derived in terms of the LFDAs for $$K_1$$ $K_1$ mesons. Using these form factors and decay constant values, the differential branching ratios of $$B\rightarrow K_1 (1270, 1400) \ell ^+ \ell ^-$$ $B\to K_1(1270,1400){\ell }^{+}{\ell }^{-}$ , $$\ell =\mu ,\tau $$ $\ell =\mu ,\tau$ transitions are plotted with respect to the four-momentum transfer squared, $$q^2$$ $q^2$ . In addition, the branching ratio values of these decays and the non-leptonic $$B \rightarrow K_1(1270, 1400) \gamma $$ $B\to K_1(1270,1400)\gamma$ decays are estimated. A comparison is made between our results for the branching ratios of $$B\rightarrow K_1(1270, 1400) \gamma $$ $B\to K_1(1270,1400)\gamma$ 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 $$q^2$$ $q^2$ 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).