In energy-constrained Low Earth Orbit (LEO) satellite constellations, in order to prolong the network lifetime, more traffic should be carried by the satellites with high battery level, which, in turn, may result in congestion in such satellites. To strike a balance, we study the multi-path routing problem which aims at Maximizing network Lifetime while maintaining a Guaranteed network Congestion level (MLGC). Particularly, we formulate such a problem as a linear programming. However, it is time-consuming that solving the proposed MLGC needs to joint multiple time intervals. Therefore, we further design an Energy Aware Multi-path Routing (EAMR) strategy without solving the optimization problem. Simulation results show that the performance of EAMR is comparable with MLGC and moreover, compared with available routing strategies, the network lifetime can be effectively improved while the required congestion level being guaranteed by implementing our proposed schemes.