Since Lily is so active maybe you could take her on some extended walks to help run off some of that energy? Ideally early in the morning and later in the evening would be helpful. If you don't like to walk it sounds like some really active games of fetch would interest her. Anything to run off the energy.
There is some info on YT in the training section on how to retrain a puppy mill dog. It sounds like this one has some of the same issues even though she is not from a puppy mill.
http://www.yorkietalk.com/forums/gen...nt-advice.html
Maybe you can find some helpful ideas there. There are also helpful tips through out the "sticky" info that is at the top of each topic area of YT.
Also it may be helpful to get a good sized crate to put Lily in for short periods of time in order to give your other dog a break or you could try tethering one of the dogs to you so that you have total control of what is happening with them. Sophie should not have to put up with Lily constantly bothering her. Lily needs to understand "no" "sit" and "stay." It takes some real patience to retrain a dog that has been abused but you cannot let her dominate your Sophie to the point that she is so unhappy. I hope you can help this poor creature that has been the victim of some ignorant humans that do not deserve to have a dog.