I am still to this day (nearly 25) terrified of shots or blood draws. Kinda sucks cause I have/had hypothyroidism which requires a lot of blood draws. I am getting better about the blood draws but still have major issues with injections. Here are some thoughts from the other perspective

1) Yes, it hurts. Don't tell me "it won't hurt" or "it will only hurt for a second". DON'T lie to me. I don't care if it doesn't hurt you, it *does* hurt me, and I hate having people trivialise my feelings. Please do not be afraid to acknowledge to your daughter that it will likely hurt, but that you will be there for her before, during, after and you will do whatever you can to help. One thing that helps - it is impossible for me to really relax, which always makes injections hurt more - have her squeeze really tightly on a stressball or wad of paper towelling or something. Then just before the injection (you will need to coordinate in advance w/the doctor and your daughter) have her let go. This will release the tension in her muscle and "force" her to relax. It doesn't eliminate the injection pain but it helps prevent soreness afterward. Giving advil or tylenol beforehand can help with this too.
I never had a numbing spray but for me that wouldn't do anything, I hate that numb feeling you get from it!!
Another thing is that one time the doctor tried using a butterfly needle for an injection on the premise that it was smaller and therefore less painful. I appreciated the thought but unfortunately it just means that it takes longer to inject all of the stuff, so all in all I prefer a regular size needle.
2) Have something ready for after. I don't normally condone bribes but this is one instance when I think it's great. I always got four stickers (one for me and one for my brother, mom, and dad) after a shot from the doctor's office, they usually only gave out one. Then I would get to go to Wendy's and get a Frosty. This was a big deal for me as we *never* had fast food growing up

I am sure you could come up with an equivalent "treat". It just gave me something else to focus on and start to calm down and let go of the trauma. If you could come up with something to have at the dr's office that would be good too.
3) Discuss with your daughter whether she wants to know in advance or not. Maybe in the future you can not tell her she will be having her appointment. You would have to arrange this with the DR apparently. For me I did better not knowing - I wouldn't sleep for days before the appointment. So instead, Mom would make the appt and not tell me until it was time to go. I had less time to freak out about it. At the same time she gradually worked with me to prepare in general so I could start to learn coping techniques.
If it does not improve and she continues to be afraid, please don't belittle her or make her feel ashamed. Needle fear is a well known phenomenon and very common. It helped me to realise how common it actually is to not feel so stupid. I had a nurse yell at me once (I was ~9) and tell me to stop acting like a baby. That did not help. There are some hypnosis programs out there that can help. I haven't gone that route yet. I just no longer have injections

There's nothing I need injected into my body at this point so I skip that. I have not had an injection since I was 12 and kicked a nurse and walked out of a drs office when they tried to give me one. I do recognise that blood draws are important so I am working on that but like I said - my issues are twofold. The pain is one, the other is the thought of something being injected into my body. I have not been able to overcome that second part yet and I am not ready to try.
One thing that we did do to help me learn to cope with blood draws was to give me an anti-anxiety medication before my blood draw, a couple of times in a row. Just enough to help me relax and calm down and not get so worked up, enough to help me focus on my breathing and get through the experience a few times. Now that I have done that, and know what I need to do, and know what to expect, etc, it is easier for me to do it on my own. Don't know if that's something to consider or not but it helped me. I also have to lie down fully every time or I will pass out.
I don't know if any of this helps - I hope something is useful and that you are able to find the best way to be supportive and help your daughter through this. Your daughter will probably be able to tell you what she needs/wants from you at least on some level. Having a plan always helped me.
MTA: I HATE the "distraction" technique. I cannot watch, but I need to be told what is happening. I hate phlebotomists that just start swabbing and stabbing. TELL ME WHAT YOU ARE DOING :P I have a "spiel" I give before every blood draw and that's part of it but I think they zone out after the "I don't do well with this and I need to lie down" and don't listen to the "please tell me what you are doing as you are doing it" bit :-/
MTA Again: Another thing the doctors tried when I was 12 or so was to give me a little needle to take home with me. The idea being mom could show it to me, I could "play" with it and get familiar with it so I wouldn't be so scared by it. That didn't work. I felt nauseous and sometimes threw up just looking at the darn thing. Nice idea, but no go for me.