In this paper, we introduce the first method that (1) can complete kernel matrices with completely missing rows and columns as opposed to individual missing kernel values, with help of information from other incomplete kernel matrices. Moreover, (2) the method does not require any of the kernels to be complete a priori, and (3) can tackle non-linear kernels. The kernel completion is done by finding, from the set of available incomplete kernels, an appropriate set of related kernels for each missing entry. These aspects are necessary in practical applications such as integrating legacy data sets, learning under sensor failures and learning when measurements are costly for some of the views. The proposed approach predicts missing rows by modelling both within-view and between-view relationships among kernel values. For within-view learning, we propose a new kernel approximation that generalizes and improves Nyström approximation. We show, both on simulated data and real case studies, that the proposed method outperforms existing techniques in the settings where they are available, and extends applicability to new settings.