Poor thing! I was diagnosed at 4 years old...and my Mum still tells me teh story of having to pin me down. As has been said, I did get used o it in the end. I think what worked with me was...
bribery! If you are a good girl and do your injection, you can go and stay with your friends, go swimming etc.
Not watching the injection. They had me practice on an orange. I started doing my own at about 6 (ish..it is a long time ago alas!). I struggled then, and I am still screamish about watching the needle go into the skin. I still sometimes do my injections through my clothes. Not really recommended, as it isn't terribly hygienic, but it meant that I was able to overcome my fear a little bit.
The other thingthat helped a lot was on a trip t the hospital, I was introduced to other kids who were in the same situation, and peer pressure worked a treat! Can your hospital help with this? There is nothing like meeting a younger kid, who is happily getting on with the injections, to make an older child come round to the idea a bit!
Sorry if the above is no help. Alas, there is no magic answer alas. Everyone is different, so all I can tell you is what worked for me. It is difficult to express teh fear that your niece is feeling..but she has my sympathy.
Good luck!