Well, it is obvious where the attitude comes from. Even when siblings are not twins, it is very tough not to compare oneself to the other. You basically have two (somewhat obvious) options :
1. Drop the student.
I have found that dealing with situations that seem sticky, having it be as "clinical" as possible is best. I don't mean heartless, but treating it as though you work for a company who has rules that you do not control. These rules apply to everyone in the studio at all times, and therefore both girls are subject to the same rules.
One suggestion for what the concept would be is as follows :
"After every such and such amount of time, I take a rather extensive amount of time to review my own potency in teaching each student. I make this effort because everyone involved is investing themselves, and the adults are investing money, toward the education we are working on in the studio. I give these assessments because I would like to be sure that each student is getting what they truly need. If, after careful review, I have found that my studio is not suitable for the individual, I will let the client know. As each student is indeed an individual, sometimes what works for one does not for another. In the event that I feel it is right to let go of a student, I will provide names of other prospective teachers, to give the student an opportunity to continue his or her education in an environment that is perhaps more suitable to the individual (please see attachment)."
Something like that anyway. And then, give each situation an actual review that always puts the child's needs first (it is ultimately what really matters and ultimately what the parents can understand best) -- which may require some time and creativity on your part, but putting this time in will serve you very well in the future, whereas putting the time and energy into keeping the student will perhaps not. In the profile of the student you wish to drop, make it very clear that your studio is not the right place to suit her needs, while it is the perfect place to suit the other's needs.
An example of something along these lines is as follows :
"I dedicate an average of ____ amount of time during off hours of teaching toward each student's individual education. I have found that putting in this time is required for each student, in order to serve each student to my best of abilities. To better serve your daughter, I would need to spend considerably larger amounts of time in preparation (this is very true, more books would need to be read, you would need to perhaps concentrate on specialized subjects... etc), which my studio simply does not allow for at this time."
This way, you are not selling yourself short, they are not excused of thier behavior (both the parents and the child), the student's needs come first, and it is clear that you cannot offer what you feel they need. You have already talked with mom, so she is aware of there being special needs.
The worst case scenario is that you lose both of them, but keep your sanity. And while you may wish to keep the one, your studio will not fold because of this and you will still feel better. If something like this does not suit you, then you have option # 2
2. Keep the student and chisel out some changes.
Since the attitude problems seem to be arising from sibling rivalry, and are probably somehow fed into at home by the parents (whether intentional or not -- and I don't think they would EVER want to hear it, so don't "talk" with them about it), this issue would need to be addressed by you in your lesson planning for each student. Ideally, the environment itself that you are teaching in would need to play a part in there being a change. Since you are teaching in their home, perhaps one of the hot-beds for the problems, you would need to be more creative in helping to create an environment suitable to each student. Some of this will have to do with something as simple as where the piano is located, who is around during the lesson and whether they are making comments or judgements. There is A LOT invovled with this option.
If this is the option you choose, expect it to be lots of work for you, and, the cold and hard reality of it is, it may not change the situation for the child. However, if done in the right way, it will be work that makes you a better teacher, and it will force you to grow. But, if this distracts you from your teaching with other individuals, it might not be worth it. And since you do have a good number of other students, I don't think the distraction would be a good idea.
Either way, give yourself something within your policies to fall back on and then work out from there.
m1469