I want her to know the truth. She is not aware that this is the truth. Will this give her some answers and make her realise she has serious issues to work on? I hope to achieve that but I don't know if it will.
If you read around on these boards, you'll find that it won't achieve your desired goal.
Hopefully she will realise she has issues and agree to engage with me about what happened.
As you said, she's blaming you for everything. There isn't any reasonable expectation that your letter will cause her to have a

moment and come around to your way of thinking. To the contrary, it will probably just upset her more.
She can't go through life repeating these cycles and hurting more people in the process. I want to convince her she needs help but I need to hook her and get her to the table first - how to I engage with her?
Why can't she go through life like that? It's worked so far for her. Your telling her that she can't do it will only cause her to walk away and find someone else who will allow her to do it for a while and then move on to the next person.
If I were you, I'd let go of the idea that you're going to convince her that she needs help. Now, if you can establish a loving and caring relationship, over time you might be able to gently guide her into getting help, but that's a long-term goal, not a first letter one.