I don't think Epsom Salt cured lily. My best guess is that the coconut oil she applied after getting them wet is what cured her. I don't think that it is a cure either. It could be the best method to manage your condition.
I tried Epsom Salt soak for 12 days but I didn't see any improvement. Only thing that helped was the oil that I applied just after getting lips wet. The oil might be helping the lips retain moisture longer which in turn helps in healing the lips. I am speaking from experience and currently I am just applying oil after soaking the lips in water for few minutes. Lips look better but I don't want to jump the gun and say that it is the cure. I will update here after few days.
If you can please post pictures of your lips, it would be really helpful.