Hi, I'd like to help you with your question.
It sounds like your sister may have a personality disorder, possibly Borderline disorder. It is very hard to deal with someone with a personality disorder. Borderline personality disorder
is a particularly difficult personality disorder because of it's symptoms and the behavior of the person. Personality disorders are a category of disorders that are difficult to treat with therapy, even if the person wants help. Everything about a person with Borderline is centered around their needs and feelings. However, if she also has depression or Bipolar, those disorders are easier to treat with therapy and medication.
There are two ways you can address your sister. One, you can lower your expectations. If she is being difficult, you can walk away. Keep in mind that reacting to what your sister is saying or doing only makes her want to do it more because she wants a reaction out of you and those around her. Leaving when she is trying to get attention can help her see that she won't get "rewarded" when she acts that way.
Also, try to keep your emotions out of your dealings with her. People with personality disorders often provoke emotional reacts in others because their behavior is so difficult to cope with. Training yourself to not react and to remain neutral can help. Develop a few non emotional responses you can provide to just about anything she says to you, especially when she tries to provoke you. "I'm sorry you feel that way" is a good response to just about any comment. It leaves your feelings out of it, negates a reaction and keeps her from feeling offended.
Getting someone to get treatment when they refuse is always difficult, especially when dealing with an adult that you have no control over. You want them to see the importance of taking care of themselves but for some reason, they refuse. It can be upsetting when you care about the person and they will not listen.
It may help to ask your sister what she might be willing to do instead. If you can, talk to her again and let her know you are upset because you are concerned about her. Ask her how you can help.
Also, try asking her if she is willing to see her doctor. Sometimes a person who won't see a therapist will see a doctor. If she is willing to go, contact her doctor ahead of time and let him/her know what you are seeing with your sister. The doctor may not be able to talk with you because of confidentiality, but you can still talk with them.
Here are some other resources to help you with ideas on how to help your sister:
I Am Not Sick I Don't Need Help: How to Help Someone with Mental Illness Accept Treatment-Xavier Amador.
The Essential Family Guide to Borderline Personality Disorder: New Tools and Techniques to Stop Walking on Eggshells by Randi Kreger
I Hate You--Don't Leave Me: Understanding the Borderline Personality by Jerold J. Kreisman and Hal Straus
I hope this helps you,