We have studied the $\tau^-\to K^-\eta^{(\prime)}\nu_\tau$ decays within Chiral Perturbation Theory including resonances as explicit degrees of freedom. We have considered three different form factors according to treatment of final-state interactions. In increasing degree of soundness: Breit-Wigner, exponential resummation and dispersive representation. We find that although the first one fails in accounting for the data on the $K\eta$ mode, the other two approaches provide good fits to them which are sensitive to the $K^\star(1410)$ pole parameters, that are determined to be $M_{K^{\star\prime}}\,=\,(1330^{+27}{-41})$ MeV and $\Gamma{K^{\star\prime}}\,=\,(217^{+68}{-122})$ MeV. These values are competitive with the standard determination from $\tau^-\to(K\pi)^-\nu\tau$ decays. The corresponding predictions for the $\tau^-\to K^-\eta^{\prime}\nu_\tau$ channel respect the current upper bound and hint to the discovery of this decay mode in the near future.