We describe the application of high-aspect-ratio ferromagnetic nanowires as microrheological probes of wormlike micelle solutions composed of equimolar cetylpyridinium chloride/sodium salicylate (CPCl/NaSal). Employing high-speed video microscopy to track the rotation of suspended nanowires in response to external magnetic fields we access both the linear and nonlinear rheology of the fluid. The linear viscosity at low rotation rates is strongly temperature dependent as expected from macroscopic rheometry. At high rotation rates the viscosity exhibits pronounced shear thinning that is surprisingly independent of temperature. The onset of the nonlinear response is characterized by a shear thickening that has no apparent counterpart in the macroscopic rheometry. Measurements involving step changes in rotation rate reveal that, once the fluid has been prepared into a shear-induced state, it exhibits nonlinear behavior far within the expected linear regime. Further, the shear-thinned fluid generates an out-of-plane torque on the wire whose magnitude varies as a power law with the rotation rate, with a power-law exponent of approximately 0.4. From time-resolved measurements tracking the motion of the wire in response to this torque we extract an effective viscosity for drag perpendicular to the nonlinear shear flow. This viscosity reveals insight into the anisotropic behavior of the shear induced state.