Ngaiz you are definitely not alone. I went to sleep school and if anything it got us in a routine which has helped heaps. I was BF'ing dd every 2 hours till I went to sleep school and she was about 5 months old.. She's now BF'ing every 4 hours. They got me feeding her 3 meals a day to make sure she wasn't hungry as I was bfing her heaps overnight. I was going to ask if you have tried formula as the last feed of the night? It's supposed to fill them up. 5 am is a really early start, if she doesn't resettle id bf and try and put her back down. White noise is my best friend! Dd sleeps so much better and is able to resettle with it on.. It might be worth a try to get her sleeping longer? Dd used to wake at about 6 and I'd put it on and she's sleep an extra hour or 2.
Bedlovers advice is great.