The $k$-ary $n$-cube $Q_{n}^{k}$ is one of the most popular interconnection networks engaged as the underlying topology of data center networks, on-chip networks, and parallel and distributed systems. Due to the increasing probability of faulty edges in large-scale networks and extensive applications of the Hamiltonian path, it becomes more and more critical to investigate the fault tolerability of interconnection networks when embedding the Hamiltonian path. However, since the existing edge fault models in the current literature only focus on the entire status of faulty edges while ignoring the important information in the edge dimensions, their fault tolerability is narrowed to a minimal scope. This article first proposes the concept of the partitioned fault model to achieve an exponential scale of fault tolerance. Based on this model, we put forward two novel indicators for the bipartite networks (including $Q^{k}_{n}$ with even $k$), named partition-edge fault-tolerant Hamiltonian laceability and partition-edge fault-tolerant hyper-Hamiltonian laceability. Then, we exploit these metrics to explore the existence of Hamiltonian paths and unpaired 2-disjoint path cover in $k$-ary $n$-cubes with large-scale faulty edges. Moreover, we prove that all these results are optimal in the sense that the number of edge faults tolerated has attended to the best upper bound. Our approach is the first time that can still embed a Hamiltonian path and an unpaired 2-disjoint path cover into the $k$-ary $n$-cube even if the faulty edges grow exponentially.