Op, I'm sorry you've been through such a harrowing time. Obviously myself and no one else here knows what happened at the birth, that is causing you to sue but I'd urge you to remember two things.
You were there due to necessity an after noon or morning a day, and not at nights, which means those staff were the ones who cared for your daughter 75 percent of the time, to make her able to come home to you
The nhs is cash strapped, and any money you take out of it, is treatment or support someone will not get, if it's really not about the money, I'd look at a way to get an investigation started, raise a formal complaint, and drop trying to take money from them if that's not your objective.
On the flip side, looking at what they wrote , which seems like no more than short notes, and your reaction. And the words you are using, about them sneering at you, thinking you're crap parents etc, sounds very extreme.
You say you're sitting crying. I think I would urge you to speak to your doctor as you may have depression.