Hi Hayley
What the A&E say is correct, If you feel more comfortable take your daughter to A&E, they won't mind at all...
To get through the night, you could do with a good amount of lucozade, coke etc to keep bumping her sugars up if required, and keep a good eye on her levels... Insulin normally works over a 4-5 hour period peaking around the 2 hours mark...
Just to reasure you that you are not the only person nor the last person to inject too much insulin, a common mistake is inject quick acting instead of background...
She will be OK, just needs close monitoring, to enusre that she doesn't hypo, as I already said if you are concerned then pop her up to the hospital I assure they won't mind at all.