The rapid deployment of low Earth orbit (LEO) satellite systems has propelled various space-based applications (e.g., agricultural monitoring and disaster response), which increasingly rely on advancements in deep learning (DL). However, ground stations (GS) cannot download such massive raw data for centralized training due to intermittent connectivity between satellites and GS, while the scaled-up DL models pose substantial barriers to distributed training on resource-constrained satellites. Although split learning (SL) has emerged as a promising solution to offload major training workloads to GS via model partitioning while retaining raw data on satellites, limited satellite-GS connectivity and heterogeneity of satellite resources remain substantial barriers. In this paper, we propose S-Leon, an SL framework tailored to tackle these challenges within heterogeneous LEO satellite networks. We develop a satellite early-exit model to eliminate training disruptions during non-contact periods and employ online knowledge distillation to incorporate ground knowledge, further enhancing satellite local training. Moreover, we devise a satellite model customization method that simultaneously accommodates the heterogeneous computation and communication capabilities of individual satellites. Lastly, we develop a partial model-agnostic training strategy to optimize the collaborative training effectiveness across customized satellite models. Extensive experiments with real-world LEO satellite networks demonstrate that S-Leon outperforms state-of-the-art benchmarks.