The recently proposed multiple kernel k-means with incomplete kernels (MKKM-IK) optimally integrates a group of pre-specified incomplete kernel matrices to improve clustering performance. Though it demonstrates promising performance in various applications, we observe that it does not \emph{sufficiently  consider the local structure among data and indiscriminately forces all pairwise sample similarity to equally align with their ideal similarity values}. This could make the incomplete kernels less effectively imputed, and in turn adversely affect the clustering performance. In this paper, we propose a novel localized incomplete multiple kernel k-means (LI-MKKM) algorithm to address this issue. Different from existing MKKM-IK, LI-MKKM only requires the similarity of a sample to its k-nearest neighbors to align with their ideal similarity values. This helps the clustering algorithm to focus on closer sample pairs that shall stay together and avoids involving unreliable similarity evaluation for farther sample pairs. We carefully design a three-step iterative algorithm to solve the resultant optimization problem and theoretically prove its convergence. Comprehensive experiments on eight benchmark datasets demonstrate that our algorithm significantly outperforms the state-of-the-art comparable algorithms proposed in the recent literature, verifying the advantage of considering local structure.