We dont know the situation, so its hard to tell.
We dont know what type of migraine she gets, but she could have rung the GP and if she couldnt get through speak to the nurse or practice manager.
Tbh I would also have rung the lecturer and taken their advice. Contact with lecturer puts them on notice before you miss the exam.
It gives me enough to show I had made an effort.
You seem obssessed with the appointment whereas i think its important to show you were trying to do something., even if you didnt get an appointment that day, then it reads better and also gives you leverage on getting the sick note because you cna get the surgery to make a note of it and the fact they couldnt do same day.
Im only interested in solving the problem and theres plenty you could do to make it easier and increae your chance of a note, then succssful claim for mitigating circumstances..