Grant-free non-orthogonal multiple access has been regarded as a viable approach to accommodate access for a massive number of machine-type devices with small data packets. The sporadic activation of the devices creates a multiuser setup where it is suitable to use compressed sensing in order to detect the active devices and decode their data. We consider asynchronous access of machine-type devices that send data packets of different frame sizes, leading to data length diversity. We address the composite problem of activity detection, channel estimation, and data recovery by posing it as a structured sparse recovery, having three-level sparsity caused by sporadic activity, symbol delay, and data length diversity. We approach the problem through approximate message passing with a backward propagation algorithm (AMP-BP), tailored to exploit the sparsity, and in particular the data length diversity. Moreover, we unfold the proposed AMP-BP into a model-driven network, termed learned AMP-BP (LAMP-BP), which enhances detection performance. The results show that the proposed LAMP-BP outperforms existing methods in activity detection and data recovery accuracy.