I'm sorry you are going through this it must be really difficult and frustrating. I think you both are coming at the situation from totally different view points. You think you are trying to spend quality time with her and make her feel included, but she probably feels judged and not good enough.
You say:
She has no pride in her appearance, doesn’t wear make up or takes care of her hair.
But also that as a way of bonding you:
have arranged for just me and her to do things together like shopping, going to the hairdressers
Have you tried doing things with her that she might actually be interested in? She likely knows that you disapprove of her appearance and so at that age would probably see your taking her to the hairdressers as a comment on this. This may be why "she acts sullen and barely speaks".
I'm sure I'm projecting a bit but when I was growing up my mum was obsessed with my appearance, she decided I didn't blow dry my hair "correctly" and so would try to force me sit there while she did it, she criticised my weigh, my hair, my taste in clothes and I hated it. I hated her for the way she valued the way I looked over who I was. I did well at school, I had really good friends, I joined in various activities, but he was embarrassed to be seen with me because I was fat and had bad hair. I felt constantly picked on. It made me feel shit about myself and fed into my depression and anxiety.
We have both grown up and changed, I realise now that she was vey insecure and her focus on appearance and what people might be thinking was a part of this. She has learnt to hold her tongue and now manages to not say anything, or even give me a pointed look, if I look like shit because I have been ill or just got in from working a 12 hour day.
I have also matured and got better, I am less ruled by my hormones but I have also had to spend a lot of time and money on therapy to undo the damage and to build up my self esteem. It took a while and we had a really bad relationship for a lot of my teens and early twenty's because of it.
I know you want the best for your daughter but do you think it is worth examining what sort of conversations you have with her? Could you maybe try to relax about things that aren't important on the grand scheme of things. Perhaps try to do something fun that isn't shopping, make up or hair related.