Eliza they can't insist on you staying. Imagine you were a single mum with several kids. Luckily you are not, and you can be there for LO. But plenty of women can't be, and the LOs are cared for perfectly adequately. I have a friend with twins, and she is a single mum. When one of them went in she had to leave her. Heartbreaking, but everyone survived.
I hated myself for walking out of the hospital but needed to do it as I really don't cope on no sleep and hospitals turn me into a wreck, really (happy to admit it, and that might make me a wuss, sod it). They will care for her in your absence, and you can miss a feed. Go home, have a favourite snack. If DH can replace you that's great. If not, E will be fine, looking after her is their job, as long as you let them know that you are leaving, for how long, and that you need to do it for you. Or for your other DD.
I was a wreck after a day. You are coping brilliantly. But get out, for your own good. Slob on the sofa with DD1, or on your own. Watch some TV. Have a break. A few hours of whatever - sleep, decent food, two episodes of Friends back to back, or lunch with a friend - whatever, will give you energy to keep going. And E will be far better off having an energised mum than someone as fed up of tubes and four walls as she is. 
I will get off my soap box now and fuck off to bed. After Graham Norton.