Robot failure is detrimental and disruptive, often requiring human intervention to recover. Our vision is 'fail-active' operation, allowing robots to safely complete their tasks even when damaged. Focusing on 'actuation failures', we introduce DEFT, a diffusion-based trajectory generator conditioned on the robot's current embodiment and task constraints. DEFT generalizes across failure types, supports constrained and unconstrained motions, and enables task completion under arbitrary failure. We evaluate DEFT in both simulation and real-world scenarios using a 7-DoF robotic arm. DEFT outperforms its baselines over thousands of failure conditions, achieving a 99.5% success rate for unconstrained motions versus RRT's 42.4%, and 46.4% for constrained motions versus differential IK's 30.9%. Furthermore, DEFT demonstrates robust zero-shot generalization by maintaining performance on failure conditions unseen during training. Finally, we perform real-world evaluations on two multi-step tasks, drawer manipulation and whiteboard erasing. These experiments demonstrate DEFT succeeding on tasks where classical methods fail. Our results show that DEFT achieves fail-active manipulation across arbitrary failure configurations and real-world deployments.