As the volume of data invested in statistical learning increases and concerns regarding privacy grow, the privacy leakage issue has drawn significant attention. Differential privacy has emerged as a widely accepted concept capable of mitigating privacy concerns, and numerous differentially private (DP) versions of machine learning algorithms have been developed. However, existing works on DP kernel learning algorithms have exhibited practical limitations, including scalability, restricted choice of kernels, or dependence on test data availability. We propose DP scalable kernel empirical risk minimization (ERM) algorithms and a DP kernel mean embedding (KME) release algorithm suitable for general kernels. Our approaches address the shortcomings of previous algorithms by employing Nyström methods, classical techniques in non-private scalable kernel learning. These methods provide data-dependent low-rank approximations of the kernel matrix for general kernels in a DP manner. We present excess empirical risk bounds and computational complexities for the scalable kernel DP ERM, KME algorithms, contrasting them with established methodologies. Furthermore, we develop a private data-generating algorithm capable of learning diverse kernel models. We conduct experiments to demonstrate the performance of our algorithms, comparing them with existing methods to highlight their superiority.