For me, the idea of a London ball being thrown for a sixteen year old girl is bothersome. Girls just didn't come out that young in London, and if there was a ball in honor of her, that would be a statement that she is now officially and fully out, therefore marriageable. If she wants to just start introducing her, then it would some other sort of party, as Janet points out--including other young people her age and their mothers, family and close friends. I think Miss Bingley would be plausibly invited, since she's already acquainted with her.
You could also have Lady Matlock throwing a ball just to throw a ball, and Georgiana is allowed to attend but not to dance. Or she could talk Darcy into throwing a ball, but for an entirely different reason. Georgiana could be at the ball, I think, but it couldn't be her ball, and she couldn't dance at it.
You could also have Lady Matlock throwing a ball just to throw a ball, and Georgiana is allowed to attend but not to dance. Or she could talk Darcy into throwing a ball, but for an entirely different reason. Georgiana could be at the ball, I think, but it couldn't be her ball, and she couldn't dance at it.