The Distributional Exact Diagonalization (DED) scheme is applied to the description of Kondo physics in Anderson impurity models. Using Friedel's sum rule, we show that the particle constraint ultimately, an essential ingredient of DED, imposes Fermi liquid behavior on the ensemble averaged self-energy and thus is essential for the description of Kondo physics within DED. Using Numerical Renormalization Group (NRG) calculations as a benchmark, we show that DED yields excellent spectra, both inside and outside the Kondo regime for a moderate number of bath sites. However, at finite temperatures the particle constraint has to be relaxed in order to correctly capture the temperature evolution of the spectra. The generalized DED scheme correctly describes the decreasing and ultimately the vanishing of the Kondo peak on increasing the temperature. Finally, we generalize DED to multi-orbital Anderson models, and apply the method to the two-orbital case.