The ability to sense the temperatures and power consumption of various key components of a chip is central to the operation of modern integrated circuits, such as processors. While modern chips often include a number of embedded thermal sensors, they lack the ability to sense power at fine granularity. This paper proposes a new direction to simultaneously identify the thermal models and the fine-grain power consumption of a chip from just the measurements of the thermal sensors and the total power consumption. Our identification technique is blind as it does not require design knowledge of the thermal-power model to identify the power sources. We investigate the main challenges in blind identification, which are the permutation and scaling ambiguities, and propose novel techniques to resolve these ambiguities. We implement our technique and apply it in three contexts. First, we implement it within a controlled simulation environment, which enables us to verify its accuracy and analyze its sensitivity to relevant issues, such as measurement noise and number of available training samples. Second, we apply it on a real multi-core CPU + GPU processor-based system, where we show the ability to identify the runtime power consumption of the individual cores using just the total power measurement and the measurements of the embedded thermal sensors under different workloads. Third, we apply it for non-invasive power sensing of chips by inverting the temperatures measured using an external infrared imaging camera. We show that our technique consistently improves the modeling and sensing accuracy of integrated circuits.