We study the charmless decays $B\to K_h\eta$ and $B\to K_h{\eta }^{\prime }$ within the framework of QCD factorization (QCDF) for $K_h=K$, $K^{*}$, $K_0^{*}(1430)$ and naive factorization for $K_h=K_2^{*}(1430)$. There are three distinct types of penguin contributions: (i) $b\to sq\overline{q}\to s{\eta }_q$, (ii) $b\to ss\overline{s}\to s{\eta }_s$, and (iii) $b\to sq\overline{q}\to q{\overline{K}}_h$, where ${\eta }_q=(u\overline{u}+d\overline{d})/\sqrt{2}$ and ${\eta }_s=s\overline{s}$. $B\to K^{(*)}{\eta }^{{(}^{\prime })}$ decays are dominated by type-II and type-III penguin contributions. The interference, constructive for $K{\eta }^{\prime }$ and $K^{*}\eta$ and destructive for $K\eta$ and $K^{*}{\eta }^{\prime }$, between type-II and type-III diagrams explains the pattern of $\Gamma (B\to K{\eta }^{\prime })\gg \Gamma (B\to K\eta )$ and $\Gamma (B\to K^{*}{\eta }^{\prime })\ll \Gamma (B\to K^{*}\eta )$. Within QCDF, the observed large rate of the $K{\eta }^{\prime }$ mode can be naturally explained without invoking flavor-singlet contributions or something exotic. The decay pattern for $B\to K_0^{*}(1430){\eta }^{{(}^{\prime })}$ decays depends on whether the scalar meson $K_0^{*}(1430)$ is an excited state of $\kappa$ or a lowest-lying $P$-wave $q\overline{q}$ state. Hence, the experimental measurements of $B\to K_0^{*}(1430){\eta }^{{(}^{\prime })}$ can be used to explore the quark structure of $K_0^{*}(1430)$. If $K_0^{*}(1430)$ is a low-lying $q\overline{q}$ bound state, we find that $K_0^{*}\eta$ has a rate slightly larger than $K_0^{*}{\eta }^{\prime }$ owing to the fact that the $\eta -{\eta }^{\prime }$ mixing angle in the ${\eta }_q$, ${\eta }_s$ flavor basis is less than 45°, in agreement with experiment. The type-III penguin diagram does not contribute to $B\to K_2^{*}{\eta }^{{(}^{\prime })}$ under the factorization hypothesis and the type-II diagram dominates. The ratio $\Gamma (B\to K_2^{*}{\eta }^{\prime })/\Gamma (B\to K_2^{*}\eta )$ is expected to be of order 2.5 as a consequence of (i) $|f_{{\eta }^{\prime }}^s|>|f_{\eta }^s|$ and (ii) a destructive (constructive) interference between type-I and type-II penguin diagrams for $K_2^{*}\eta$ ($K_2^{*}{\eta }^{\prime }$). However, the predicted rates of $B\to K_2^{*}{\eta }^{{(}^{\prime })}$ in naive factorization are too small by 1 order of magnitude and this issue remains to be resolved. There are two $K^{(*)}{\eta }^{{(}^{\prime })}$ modes in which direct $CP$ asymmetries have been measured with significance around $4\sigma$: $A_{CP}(K^{-}\eta )=-0.37\pm 0.09$ and $A_{CP}({\overline{K}}^{*0}\eta )=0.19\pm 0.05$. In QCDF, power corrections from penguin annihilation which are needed to resolve $CP$ puzzles in $K^{-}{\pi }^{+}$ and ${\pi }^{+}{\pi }^{-}$ modes will flip $A_{CP}(K^{-}\eta )$ into a wrong sign. We show that soft corrections to the color-suppressed tree amplitude $a_2$ in conjunction with the charm content of the $\eta$ will finally lead to $A_{CP}(K^{-}\eta )=-{0.15}_{-0.28}^{+0.19}$. Likewise, this power correction is needed to improve the prediction for $A_{CP}({\overline{K}}^{*0}\eta )$.

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