I feel like a bad mum

My daughter is 8 weeks old. Last night she was awake from 8:00pm-3:30am which is highly unlike her to be awake that long. She was so fussy. Just wanted to be on my boob for comfort. When she was doze off o would wait a little bit until I thought she was more asleep and then take her off my boob. But then she woke up. She wouldn’t sleep at all. I could tell she was becoming over tired. Her eyes were so red, she was fighting sleep so hard. So I caved and just gave her half a dose of Tylenol, and she finally fell asleep. I feel so bad that I had to give her medicine to sleep, and I will not make it a habit at all. Because it’s wrong. But she just wouldn’t fall asleep 😩