Soft robotic hands can facilitate human-robot interaction by allowing robots to grasp a wide range of objects safely and gently. However, their performance has been hampered by a lack of suitable sensing systems. We present a flexible and stretchable multi-modal sensor network integrated with a soft robotic hand. The design of wired sensors on a flexible metalized film was embodied through a manufacturing approach that uses both UV laser metal ablation and plastic cutting simultaneously to create sensor electrode patterns and stretchable conductive wires in a Kirigami pattern. Temperature and proximity sensors are combined into a single network. We evaluated the interconnects and sensors by measuring an impedance change to each external stimulus and showed that sensor readings are not substantially affected by stretching the network. With the sensor sheet wrapped around a soft robotic gripper, we demonstrated several interaction scenarios, including identifying hot and cold water bottles, a warm burrito for food handling, and a warm baby doll for future medical applications.