Hey. I can totally feel with you. We got a baby girl around 2 month ago. It's one of the toughtest time we had, but also one of the luckiest. Seeing my baby girl lucky and growing gives me infinite power! (kinda ;-))
With only few hours sleep I manage somehow every day to survive all the trouble that additionally somehow arrives just in time in the office.
I don't know if this helps you but let me tell you some things:
Being mother is not an easy thing. She and your baby girl need you! Do everything you can (I'm sure you already do) and don't give up. Show her that you mean it. Write your feelings down in a card. Buy flowers. Look that she can take a break sometimes. She needs breaks and she needs help. Go walking with the baby girl in the evening. Call people near to you to help out a bit sometimes at home if they can (e.g. mother). Maybe you can take a day off from time to time or at least come home ealier. And sometimes you'll just need a cold beer when everyone is asleep late in the evening.
Sharing this story is a good step of you as even telling someone can be a relief.
J