If she literally fell asleep as soon as she put it there, then in the immediate term, I'd let her sleep - no fight was ever won when everyone is exhausted or just woken up to be reprimanded. But I'd take the sign off her door and put it to one side.
Once everyone has calmed down, slept, showered and fed or whatever needs to be done to get back on an even keel, I'd be sitting her down and telling her that the language is unacceptable - end of, no debate, and this is the punishment for that - keep it in line with what normal punishments are in your family. (BTW, you and DH might need to have a serious think about the household language; it doesn't necessarily come from YouTube. I love a good swear, but that comes with consequences you need to parent to)
But I wouldn't get into the detail of her hating you in that particular moment; every kid has hated their parent and their rules at some point. And if you kick off about everything, including screen time, the sign and the swearing, it all just becomes a big mess of being told off, and the points won't hit home.
But then you need to really deal with the screen issue because that sounds like the underlying problem - so new and stricter controls, new and stricter time limits. And you and DH need to stick to those and follow through when she inevitably kicks off or when things are tough, and a screen might be an "easy" option. Nope, set the rules and stick to them.