Well done on not reacting to your dd and keeping things calm. That sounds very difficult.
From everything you’ve written, I’d say neurodivergence. Girls mask and don’t present in a classic way so reading online about traits won’t necessarily help.
Have you heard of PDA - pathological demand avoidance? The lying will be doing something for your dd. A way to avoid. I doubt she’s deriving Machiavellian pleasure from this. More likely she’s found this is a neat way to manage her anxiety and isn’t totally understanding the social implication, which needs explaining to her. She’s smart. It’s effective. I suggest looking at ASD compulsive lying for more information.
The inability to cope with when you were ill imo was out of a place of abject fear. Meaning your dd literally shut down because she was too afraid you might die as well. She is heavily avoidant and literally can’t talk about it because it’s too difficult to process. My dd did exactly the same with me with my second surgery. I thought she coped with the first but she obviously didn’t. She was then really cruel to me with the third. I now know she disengaged because it was too painful to do otherwise because my first surgery was so traumatic for her and she was afraid I was going to die. If she cut me off she didn’t need to care about me and in her head my death would hurt her less.
If you suspect an eating disorder your dd has ‘grown out of’, please be careful with this. It was a real struggle to get my dd to eat a variety of foods right from a baby. But I did it. Your dd is very anxious and it’s probably going to get worse. At the start of year 11, my dd developed eating issues. Looking back the signs were there but I thought things would ease post GCSE and I lost loads of weight at her age as well. My dd ended up with anorexia and the ARFID came back, big time. She isn’t diagnosed because she refused to engage with CAMHS. We are doing this privately and the eating disorder coach said anorexia jumped on the back of the ARFID.
My dd is in therapy. It took many months to engage. She refused to go for a long time and can now withstand a 1.5 hour appointment once a month. For us, the rest of the time, it’s twice weekly parent sessions, so you can see the lion’s share to bring dd along happens outside of her therapy.
From everything you’ve written your dd isn’t just going to engage in therapy. You would need a therapist, who specialises in neurodivergence. And that would start with you getting the tools yourself to be able to deal with your dd in a different way so that you can get her to buy into therapy. This is how I got my dd to therapy. And a dynamic therapist, who doesn’t sit there waiting for your dd to engage but actively teaches you how to get her to work with the therapist.
If you really want the room sorted, I would suggest that you give her the tools. Give her a laundry basket for her dirty clothes that she brings down on a Monday and Thursday for example. If there’s room, consider buying a free standing towel hanger. Instigate rules around eating. Eg cups and plates to be brought down every morning before school. And include cleaning her room. Make a list of things you want to see happening.
I imagine you’ll know not to start with the list. This is the sort of thing that would work with my dd if she had messy bedroom issues. Get your dd’s buy in by saying that you want to respect her privacy, however, you realise that because you were so eager to maintain this that you’ve forgotten she’s going to be an adult one day and you realise you haven’t been teaching her the skills she is going to need to look after herself. And that this is on you. And that these are things for her and her future. That you're willing to help show her how to do stuff and you don’t know what that looks likes. Maybe specific tasks or setting aside a day, when the two of you could sort her room out for her. Ask her what she thinks. Then present her with the list of things. Don’t make it too hard for her.
If she goes mad / refuses, just explain that you want to continue to respect her privacy and this is the way to do that. However, if this is not something she wants to do, as the adult, you're going to come into her room to sort it for her or with her eg every Sunday. So it’s in her best interest to do this so that she can maintain her privacy.
Then put the list on the fridge or a cupboard so she sees it all the time.