The development of simple and fast hypergraph spectral methods has been hindered by the lack of numerical algorithms for simulating heat diffusions and computing fundamental objects, such as Personalized PageRank vectors, over hypergraphs. In this paper, we overcome this challenge by designing two novel algorithmic primitives. The first is a simple, easy-to-compute discrete-time heat diffusion that enjoys the same favorable properties as the discrete-time heat diffusion over graphs. This diffusion can be directly applied to speed up existing hypergraph partitioning algorithms. Our second contribution is the novel application of mirror descent to compute resolvents of non-differentiable squared norms, which we believe to be of independent interest beyond hypergraph problems. Based on this new primitive, we derive the first nearly-linear-time algorithm that simulates the discrete-time heat diffusion to approximately compute resolvents of the hypergraph Laplacian operator, which include Personalized PageRank vectors and solutions to the hypergraph analogue of Laplacian systems. Our algorithm runs in time that is linear in the size of the hypergraph and inversely proportional to the hypergraph spectral gap $λ_G$, matching the complexity of analogous diffusion-based algorithms for the graph version of the problem.