Your sister is lucky indeed to have such a caring family. Many people would have given up long ago. Unfortunately, her moods and drama have dominated your family so long that you are all exhausted. She has come to expect that everyone will give her unlimited attention whenever she demands it. It hasn’t helped her. It has cost everyone else a great deal.
For that reason, I suggest you stop talking to your sister about what she should do. You already know that she’ll continue to procrastinate in spite of your best efforts to get her to a psychologist. You already know that she isn’t motivated to help herself as she should. Do yourself a favor and drop the argument. It’s only frustrating you.
Instead, I think you and your parents should get into family therapy. You all deserve to live your own lives but I know you won’t abandon her. You need some help to find a way out of the current situation that will also provide her some support. An experienced family therapist can help you all find new ways to manage your sister and better ways to take care of yourselves. Please understand that I’m not saying that you and your parents have done anything wrong or that there is anything wrong with you. I am simply suggesting that when good people have run out of options, it is helpful to get new eyes on the situation.
I wish you well.