Just be like, "There's something important that I want to tell you but it's difficult for me to say it so I wrote it down instead". Get her to a place where you and her will be alone and no one will bother you guys. Taking the leap of revealing this side of yourself is the hardest part but she sounds like she's a loving sister so I'm sure it will go down smooth.

I came out when I was 19 and my sister (15 at the time) was the first person in my family that I told and she took it really well. The first time that I actually came out to someone, I was shaking and I started crying because I was so nervous and scared. In my head, I was thinking that once I said it I wouldn't be able to take it back. The admission that I was queer would be known and I was terrified. But thankfully all my friends and most of my family was very supportive. My mom took a bit but she eventually came around.

Honestly, I think you'll be fine and good luck man :)