Having no medical experience whatsoever, I still think you did the right thing. I am a mum, and I once watched an experienced paeds doctor take 12 minutes to draw blood from my perfectly healthy infant, who was a red-faced, snottery, squalling mess by the time he'd finished. In fact, it was a contest as to who was the reddest, her or the doctor! What happened after you delivered her to A&E is their responsibility, although I do think it reflects on your dedication that you were so concerned at the delay. Take a deep breath, and repeat after me, "I did the right thing". Keep repeating it until you convince yourself.