An open challenge in soft matter science is the ability to create hydrogels that are soft but also have high fracture energy. A possible solution to this stiffness-toughness conflict has arisen through a class of material known as hybrid double-network hydrogels, which combine two polymeric networks with diametrically opposed chain stiffness and with both covalent and physical crosslinking. The vast parameter space inherent to such a system means it is difficult to identify the precise compositional parameters that lead to both high toughness and low stiffness. In this work, we address this challenge through a Design of Experiments (DoE) framework used to establish the statistical relationship between factors and mechanical properties of a hybrid double-network hydrogel. The crosslinking density of the networks is noted to play a prominent role in determining the stiffness of the hydrogel, while the network characteristics of the ductile network determine the toughness of the hydrogel. We also report that contrary to observations in current literature, it is possible to toughen the hydrogel without stiffening it. Therefore, the present experimentation and optimization exercise provides a hands-on guide for the use of DoE to determine the conditions for optimised mechanical properties of thin hybrid double-network hydrogels for various applications.