Kernel methods provide an efficient nonparametric model to produce adaptive nonlinear filtering (ANF) algorithms. However, in practical applications, standard squared error based kernel methods suffer from two main issues: (1) a constant step size is used, which degrades the algorithm performance in non-stationary environment, and (2) additive noises are assumed to follow Gaussian distribution, while in practice the noises are generally non-Gaussian and follow other statistical distributions. To address these two issues simultaneously, this paper proposes a novel kernel normalized mixed-norm (KNMN) algorithm. Compared to the standard squared error based kernel methods, the KNMN algorithm extends the linear mixed-norm adaptive filtering algorithms to Reproducing Kernel Hilbert Space (RKHS) and introduces a normalized step size as well as adaptive mixing parameter. We also conduct the mean square convergence analysis and demonstrate the desirable performance of the KNMN algorithm in solving the system identification problem.