This paper is concerned with the problem of reliable $H_{infty}$ filter design for a class of mixed-delay Markovian jump systems with stochastic nonlinearities and multiplicative noises. The mixed delays comprise both discrete time-varying delay and distributed delay. The stochastic nonlinearities in the form of statistical means cover several well-studied nonlinear functions. And the multiplicative disturbances are in the form of a scalar Gaussian white noise with unit variance. Furthermore, the failures of sensors are quantified by a variable varying in a given interval. A filter is designed to guarantee that the dynamics of the estimation error is asymptotically mean-square stable. Sufficient conditions for the existence of such a filter are obtained by using a new Lyapunov-Krasovskii functional and delay-partitioning method. Then a linear matrix inequality (LMI) approach for designing such a reliable $H_{infty}$ filter is presented. Finally, the effectiveness of the proposed approach is demonstrated by a numerical example.