We investigate the mechanical properties of a magnetic temperature-sensitive hydrogel at varying concentrations of covalent and physical cross-linking. The hydrogel consists of covalently cross-linked poly(N-isopropylacrylamide) (PNIPAAm), physically interacting nanoclay particles, and magnetic ferric oxide nanoparticles. The physical nanoclay network exhibits strong viscoplastic behavior, and we find that increasing nanoclay content improves both strength and toughness in the double network materials. We investigate the behavior of the gels using a nonlinear viscoplasticity model with a modified rule of mixtures approach and attribute the observed trends to two factors: (a) the yield-stress behavior of the nanoclay network and (b) load-sharing interactions between the PNIPAAm and the nanoclay. Our findings indicate a strong correlation between the mass ratio of covalent cross-linker used and fractional percolation of the PNIPAAm network.