On the injections, my son was very traumatised by the blood draws and cannula for the drip at diagnosis. The dehydration and the effect of the condition pre diagnosis means that it is more difficult for the nurses to find a good vein. He was able to start injecting himself while in hospital (age 8) and that element of control helped, but larger things like pump or cgm insertions and blood draws (not finger pricks) were very difficult for him to start off with.
As your LO is so little, she won't be able to inject herself yet but as NachoAddict says, maybe start thinking about different ways to involve her - e.g. putting the test strip in the meter, pressing the button on the lancing device (finger pricker), counting with you when doing the injection (we used to count in different languages). She can try to guess what number she is going to be on the BG meter. There is a thing called a buzzy buzzy4shots.co.uk which can help distract some children.
Also, I'd say treat this like potty training - sticker charts, a favourite book or game on the phone while the injection is happening, a chocolate button afterwards. I know you will be worried that this will set you up for problems with her expecting a reward for ever, but that's not my experience. You just need to do whatever you need to do to get through this, so don't make it any harder on yourself.
My lovely online friend was diagnosed at about the same age as your daughter, more than 60 years ago, when the tools that we have now to manage the condition just didn't exist. In the past few years he has been climbing Mount Kilimanjaro and trekking through Peru which is impressive for someone in their 60s, let alone someone with type one. Prognosis was much worse in those days. This is his story. www.diabetes.org.uk/your-stories/type-1/ive-lived-with-type-1-diabetes-for-60-years
Also you might want to check out Roddy Riddle who does all sorts of extreme endurance events. My son interviewed him earlier this year vimeo.com/214374612
Please do PM me if you would like and I would definitely recommend the facebook sites for support. Huge hugs.