This work explores non-prehensile manipulation (NPM) and whole-body interaction as strategies for enabling robotic manipulators to conduct manipulation tasks despite experiencing locked multi-joint (LMJ) failures. LMJs are critical system faults where two or more joints become inoperable; they impose constraints on the robot's configuration and control spaces, consequently limiting the capability and reach of a prehensile-only approach. This approach involves three components: i) modeling the failure-constrained workspace of the robot, ii) generating a kinodynamic map of NPM actions within this workspace, and iii) a manipulation action planner that uses a sim-in-the-loop approach to select the best actions to take from the kinodynamic map. The experimental evaluation shows that our approach can increase the failure-constrained reachable area in LMJ cases by 79%. Further, it demonstrates the ability to complete real-world manipulation with up to 88.9% success when the end-effector is unusable and up to 100% success when it is usable.