This paper addresses the problem of activity detection in distributed Internet of Things (IoT) networks, where devices employ asynchronous transmissions with heterogeneous power levels to report their local observations. The system leverages an intelligent reflecting surface (IRS) to enhance detection reliability, with optional incorporation of a direct line-of-sight (LoS) path. We formulate the detection problem as a binary hypothesis test and develop four detectors: an optimal detector alongside three computationally efficient detectors designed for practical scenarios with different levels of prior knowledge about noise variance, channel state information, and device transmit powers. For each detector, we derive closed-form expressions for both detection and false alarm probabilities, establishing theoretical performance benchmarks. Extensive simulations validate our analytical results and systematically evaluate the impact of key system parameters including the number of antennas, samples, users, and IRS elements on detection performance. The proposed framework effectively bridges theoretical optimality with implementation practicality, providing a scalable solution for IRS-assisted IoT networks in emerging 6G systems.