We present a systematic study of thermodynamics, structure, and rheology of mixtures of cationic wormlike micelles and like-charged nanoparticles. Structural and thermodynamic measurements in dilute surfactant-nanoparticle mixtures show the formation of micelle-nanoparticle junctions that act as physical cross-links between micelles. The presence of these junctions is shown to build significant viscosity and viscoelasticity in dilute and semidilute WLMs, even in cases where the fluid is Newtonian in the absence of nanoparticles. Increases in viscosity, shear modulus, and relaxation time, as well as decreases in entanglement concentration, are observed with increasing particle concentration. As such, nanoparticle addition gives rise to a so-called "double network" comprised of micellar entanglements and particle junctions. A simple model for such networks is proposed, where the elasticity can be tuned through two energetic scales, the micellar end-cap energy and micelle-nanoparticle adsorption energy. As a practical application, the results demonstrate that nanoparticle addition provides formulators a unique method to tailor surfactant solution rheology over a wide range of conditions.