Cowapjn ... like I said before, this internet thing isn't helping you now. You are frustrated, you feel helpless and unsupported. Constantly rehashing the same discussions with each of us is just reinforcing those feelings and bedding them in deeper and deeper.
My guess is that just now, neither of you nor your ex has a clear head. Even the friendliest break ups I've seen others go through have had the effect of making both partners magnify the faults of the other.
You are (probably) both being unreasonable. But the more that you focus on how unfortunate it is that this is the happy and uncomplicated situation you'd wish for, the more of a wedge you'll drive between you and your ex. Both in your mind and hers.
Our circumstances are very different but I can only talk from my experience. I am in a happy relationship, with the usual ups and downs (and the thankfully now dead father in law I was talking about before), so I was there when my first was born and when she and my partner came home, I drove. The hospital was about two miles away ... the journey took about half an hour! I thought everyone else on the roads was a crazy. They were going so fast! Didn't they realise they could kill someone? The first month or so, I lived in terror that somebody was going to come and say there had been a mistake and they had to take the baby away, she was the wrong one, too perfect. It was over a week before we left the house, and then we got the shakes after about five minutes and had to go home with the fears.
Now ... does that sound sane to you? I hope not. And I'm the dad ... I didn't have the hormonal adjustments and all that jazz.
What I'm getting at is that you could be a saint your home as child-safe, padded and secure as any creche ... and your ex might still be saying "no way, not good enough for my girl". ... In the actual real world it isn't that great. You've split. The trust there was will need to be rebuilt slowly and painstakingly. It's going to take time, hard work and inhuman amounts of patience. I do not envy you the hill you have to climb.
But you are showing determination. Your tenacity on this thread putting forward your story and arguing your case speaks of someone with the strength of character to see it through.
Get off the computer. Get out for a walk and clear your head. When you are calm, call and ask if you can pop round. DO NOT ASK to be allowed to take the baby out or away from her mum. Ask how you can help but don't force help on your ex. Deal with the poo, that is always appreciated.
Small steps, softly taken, will bring you to a place of happiness faster than the strides your pride in your daughter is crying out for you to take.
Just my view. I don't know you. It seems your ex's sister does, though. And it seems like she sees your pain and is fairly neutral. Talking with her will probably help more than this forum ever will. But don't try to make her be 'on your side' or you'll just cause problems betterment the sisters. Just talk. Better still, just listen.
Get off the computer (or phone or whatever). Really.
Best of luck