The cathode powder is obtained by wet crushing and screening, and the leaching behavior of Li, Ni, Co, Cu, and Al is then investigated using a ternary leaching system composed of ammonia, ammonium sulfite, and ammonium bicarbonate. Ammonium sulfite is necessary as a reductant to improve the Li, Ni, and Co leaching efficiencies, and ammonium bicarbonate acts as a buffer in ammoniacal solutions. A detailed understanding of the selective leaching process is obtained by investigating the effects of parameters such as the leaching reagent composition, leaching time (0–300 min), temperature (40–90 °C), solid-to-liquid ratio (10–50 g/L), and agitation speed (300–700 rpm). It is found that Ni and Cu could be almost fully leached out, while Al is hardly leached and Li(60.53%) and Co(80.99%) exhibit a moderate leaching efficiency. The results show that the optimum solid-liquid ratio of the leaching system is 20 g/L, and the increase of temperature and reaction time is beneficial to metal leaching. The leaching kinetics analysis shows that the chemical reaction control explains the leaching behavior of Li, Ni, and Co well. Therefore, this work may be beneficial for further recycling valuable metals from leaching solutions by introducing an extraction agent.