It seems like 'my sister and her' are both subjects so it should be 'my sister and she'... And it would be if they worked on something specific together (the obvious object). Eg: My sister and she work on this project.
But the "together" at the end changes everything right? Unbundling the sentence it is clear that it is actually saying "my sister works with her", making 'her' the object. In this case should it be my sister and her work together? Does the together really change the object?