What an awful situation, I really feel for your daughter - sounds terrifying. Regarding gaining good control, there a some rules that I follow.
Testing
The most important part of control. Test before and after every meal, before bed and upon waking. Keep a record of this. If you need diaries, ring this number (Novo Nordisk - they make Novorapid) and they will send some to you: 0845 600 5055.
Basal dose
Also known as long acting, it's the 24 hour insulin that covers your insuln needs between meals and overnight. I use Lantus as my basal insulin and it looks like your daughter uses Levermir. It's very important that this dose is correct, without it being correct, it is very difficult to get the correct fast-acting dose without the basal being correct and her numbers will be all over the place. The way to tell is by not eating, or injecting short acting insulin at least 4 hours before bed. Then, compare the before bed test and the upon waking test. If there is a difference of more than 1.6 mmol/l, then the basal is incorrect. For each change of 1.6 mmol/l you will need to adjust the basal by 10% accordingly.
Fast-Acting dose
I use Novorapid for my fast acting insulin, this is given to cover meals. As others have mentioned, 1 unit of fast acting insulin per 10g of carbs is a good place to start.
Diet
Diet is the greatest thing that will affect sugar levels, in particular carbohydrates. If your daughter reduces the amount of carbs that she eats then she won't go as high. I don't know how reducing your daughter's carbs will affect her gastro-illness, so you will need to look into it. If all is ok on that front, then I'd suggest that cutting her carbs will make the most dramatic improvement in her sugar levels. I don't eat bread, pasta, rice, potato etc and that is how I got rid of my background retinopathy and got my HbA1c down to 6.1%. There's a low carb section on this forum, if you are interested, then there are loads of good ideas on there. But, be careful about any dramatic, sudden drops in overall blood sugar control as this will make retinopathy worse. So perhaps aim to get her insulin dose correct before aiming for radical changes in diet.
Exercise
Regular exercise gives more consistent readings and reduces the amount of insulin required. Smaller doses of insulin means that you will have smaller mistakes, both high and low. I don't know if your daughter manages to do any exercise given her complications, but I'd suggest she didn't do any for the time being as it will disrupt the calculation of basal and fast-acting insulin.
As mentioned, don't aim for a dramatic increase in good control as this will make the retinopathy worse. Her consultant will advise an HbA1c to aim for.
This forum is full of experienced, well-controlled diabetics, who in my opinion are the best resource for gaining good control. Please, please post her daily test readings and diet on here as we will be able to help.