Statistics will vary depending on where you look. Whether she will be sober or not when she leaves is completely on her shoulders. She will need to decide that she is done with drinking and that she wants a better life for herself.
I wouldn't pay for someone's rehab. There are free rehab options and as a recovered addict, I know from personal experience that if the A does not want to change, then no matter how nice the rehab is or how long the rehab, won't make a difference. It all comes down to one factor, whether she is ready to turn her life around. I don't know any addicts that got clean just because they went to rehab, they needed to be ready to change their entire lives around.
As for the letter, I would be honest. Tell her how her addiction has affected you. She is 26 years old, she can handle the truth. In fact, I think knowing the truth will give her a reality check and force her to face that her addiction isn't just ruining her life, it is affecting those around her that she loves. When I was in active addiction I would tell myself that I don't get why everyone cares that I use, it was my life and it doesn't affect them. It is when I started talking to my mother and siblings that I realized that it affected everyone. It wasn't that they just were hurt and worried by my addiction, it was that I was slowly killing them all along with myself.
Also, remember, her being in rehab doesn't mean that you need to take time out of your life or take off work. She is there and she is safe, the rest is up to her. You should take this time to work on setting boundaries and taking care of yourself. Focus on you, not her. You health (emotional and physical) should be a main focus.