In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations.