With the development of the Internet of Things (IoT), it requires better performance from wireless sensor networks (WSNs), such as larger coverage, longer lifetime, and lower latency. However, a large amount of data generated from monitoring and long-distance transmission places a heavy burden on sensor nodes with the limited battery power. For this, we investigate an unmanned aerial vehicles assisted mobile wireless sensor network (UAV-assisted WSN) to prolong the network lifetime in this paper. Specifically, we use UAVs to assist the WSN in collecting data. In the current UAV-assisted WSN, the clustering and routing schemes are determined sequentially. However, such a separate consideration might not maximize the lifetime of the whole WSN due to the mutual coupling of clustering and routing. To efficiently prolong the lifetime of the WSN, we propose an integrated clustering and routing scheme that jointly optimizes the clustering and routing together. In the whole network space, it is intractable to efficiently obtain the optimal integrated clustering and routing scheme. Therefore, we propose the Monte-Las search strategy based on Monte Carlo and Las Vegas ideas, which can generate the chain matrix to guide the algorithm to find the solution faster. Unnecessary point-to-point collection leads to long collection paths, so a triangle optimization strategy is then proposed that finds a compromise path to shorten the collection path based on the geometric distribution and energy of sensor nodes. To avoid the coverage hole caused by the death of sensor nodes, the deployment of mobile sensor nodes and the preventive mechanism design are indispensable. An emergency data transmission mechanism is further proposed to reduce the latency of collecting the latency-sensitive data due to the absence of UAVs. Compared with the existing schemes, the proposed scheme can prolong the lifetime of the UAV-assisted WSN at least by 360%, and shorten the collection path of UAVs by 56.24%.