We demonstrate the application of the transition manifold framework to the late-stage fibrillation process of the NFGAILS peptide, a amyloidogenic fragment of the human islet amyloid polypeptide (hIAPP). This framework formulates machine learning methods for the analysis of multi-scale stochastic systems from short, massively parallel molecular dynamical simulations. We identify key intermediate states and dominant pathways of the process. Furthermore, we identify the optimally timescale-preserving reaction coordinate for the dock-lock process to a fixed pre-formed fibril and show that it exhibits strong correlation with the mean native hydrogen-bond distance. These results pave the way for a comprehensive model reduction and multi-scale analysis of amyloid fibrillation processes.