An exact formulation of the dyadic Green's function in terms of new spherical vector wave functions for the electromagnetic sources placed inside of a radially multilayered sphere of reciprocal and nonreciprocal chiral media is presented according to the method of scattering superposition, and the general formulas for the determining the focal points of spherical chirolens are derived using the approximation of impedance matching and geometrical optics. Furthermore, we investigate the electromagnetic responses of inhomogeneous spherical chirolens antenna characterized by different constitutive parameters, and the effects of impedance matching as well as chirality admittance on the radiation patterns are examined in detail.