WOW! You need sleep. The best gift you can give your baby girl is her mommy at her best! She does not need to be feeding at night by this age. She needs her sleep as much as you do. She may just want to suck to get back to sleep and not want to eat at all. Try a pacifier. I did not want my kids to be dependant on pacifiers, but it is better than finger sucking because you are in control of when that habit stops. Babies are so oral, she may just need that sucking and that is the only reason she nurses at night. The fact that she doesn't feed for very long tells you that it is not nutrition that she is needing.
Also, I would encourage you to allow your baby to cry. Babies need to cry! It is good excersize and sometimes they need to get rid of extra energy they have, in order to sleep. Another gift you could give your baby is allowing her to learn how to sooth herself. I'm not suggesting that you don't go check on her etc. She may need help with her pacifier, be cold, need changing etc. However, you can tend to her needs, reasure her that she is okay and put her back down. She will sleep better by herself and you most definately will.
Good luck, I know this is a hard time. Babies are so precious and we all want what is best for them. Remember that you need to love her enough to set her up for healthy living wich is sleeping at night and waking in the morning rested. God Bless you!