When I was 15 I had to have surgery, when the doctor told me I basically didn't hear anything else that he told me and had a full blown panic attack. I remember coming out of the room absolutely indignant that I would not be having the surgery.
My DM sat me down and talked me through the process - first stage, consultation, next stage, book a date, then the surgery. She made it clear we could go at my pace (even though my surgery was very much needed and this likely wasn't the best option). She said if I could do it her and my dad would get me a treat.
First stage arrived and we met with the consultant, who told me the options re local/general anesthetic. I said I wanted general. He then booked the surgery. Panic attack number 2.
Then the date of the surgery, I was almost 16. So I get to the hospital and they tell me I'm going on the adult ward to wait, that meant I would be going in alone. Panic attack number 3. My mum stood up for me and said that I needed her there or I wouldn't go in, the nurses agreed and let her come down as far as the actual anesthesia room. I got an iPod nano as my treat - which was great while i rested from the surgery.
Following the surgery I needed a blood test. It took 6 months of therapy before I had that blood test. My mum dutifully took me to therapy every week and never said a word about how "easy" blood tests are or anything. One day I came out of therapy and said I wanted to get it done - we drove to the hospital and my mum again was my biggest advocate, she told the nurses I was afraid and the nurses talked me through it, they told me I could leave at any time and come back another day. Funnily enough, the thing that got me through the blood test - the iPod nano, the nurses told me to put on my favourite song and lay down.
I am still so fearful of surgeries and blood tests etc but a lot can be said for positive first experiences because I now cope much better (no panic attacks!) in those sorts of situations and have had many blood tests/unpleasant examinations since.
I think if I was in your shoes and talking to my daughter, I'd break it down into sections, offer to go at her pace and offer a treat. As we all know, fear is a horrible emotion, and it's hard when you're young to understand that even adults are fearful of these kinds of things!