I have a 7yr old DD and you could have been describing my DD in your post. Personally, I dont think 7.45 is too early for bed but again my DD stays awake until 9-10pm.
I have more or less accepted it now, the only thing is that she is not allowed anything after 8.30 i.e. playing games etc. She soon drops off then! I wouldnt worry too much, every now and then she will go to bed early and sleep straight away, probably her lack of sleep catching up on her.
Just think when she is a teenager she will want to get her beauty sleep!!!