This is basically what I've written:
Dear [Sis's name]-
I have written and re-written this letter both on paper and in my head hundreds of times. I'm not sure what, if anything, you know about me. In short, I am your sister. Technically, half-sister; Your dad is my biological father. I'm 33 years old and live in Ypsilanti. I grew up there and just recently moved back after spending ten years in Los Angeles, California. I am married and have a three year old daughter. She's the cutie in the picture. I'm the brunette.
I waited until you were an adult to contact you so that you could decide if you wanted pursue a relationship with me (which would be great) or not (which is also totally fine). I have always wanted to get to know you, but I also wanted to respect your family and not just jump in and throw you for a loop if you didn't know about me. I wanted to be as unintrusive as possible. You've spent your whole life without an older sister and might not want one (we can be an odd breed). If you don't, like I said, that's totally fine. I just wanted to let you know that I'm here if you do. You can write me at [address] or call me at [phone numbers].
I hope all is well with you and your family. I look forward to hearing from you.
Aimee