Hyperbolic metamaterials can enhance spontaneous emission, but the radiation-matter coupling is not optimized if the light source is placed outside such media. We demonstrate a 3-fold improvement of the Purcell factor over its outer value and a significant enlargement in bandwidth by including the emitter within a Si/Ag periodic multilayer metamaterial. To extract the plasmonic modes of the structure into the far field we implement two types of 1D grating with triangular and rectangular profile, obtaining a 10-fold radiative enhancement at visible frequencies.