To be let down by friends and boyfriend must make it extra hard for her. My daughter had the 'advantage' of not really having any friends (that's a whole other worry!) But at least it meant she wasn't particularly aware of what others were doing, although she must have some idea of contemporaries via social media.
I found it, rather selfishly, I admit, rather more depressing for me than it appeared to be for dd. Looking back though, I think we did manage to avoid total inertia in that, to begin with, dd was focusing on learning to drive and passing her test. That really helped her to get out of the house. Also, being shy like your dd, I think it helped her to have an instructor who she had to interact with. She recently told me how much she enjoyed the chats they had. It's little things like that that many might dismiss as insignificant, but I think it helped my dd's confidence. After learning to drive, she then focused on the evening classes I mentioned earlier, and some travelling/holidays/concert going etc. Again, all helped to build confidence and keep her mental health on an even keel. Maybe there is some little thing like that you could encourage her to do? I realise the job hunt is the main focus, but it sounds like she could do with working on herself a bit too?
I don't know whether it helps for you to be too involved in the application and CV side of things. When my husband tried to give advice re CVs, he was told his ideas were out of date! We were told cover letters and chasing up phone calls were not the done thing too. Basically, all our advice was, on the surface, rebuffed. I mostly left her to work it out for herself as I found her seeming indifference frustrating. It is only later that I realised she wasn't indifferent, just frustrated herself at lack of progress. I guess having parents 'nagging on' (we thought we were encouraging!) probably isnt that great for their mental health either.
Amongst other things, your dd really needs to keep plugging away at the temping agencies. She just needs to get on their radar. Could she ring them? (I know it isn't always possible and a lot of contact is made via specific job applications.) My dd is hopeless at ringing people - too shy - but once that first contact is made, it does get easier. She had several minimum wage roles offered to her once that ball was rolling. I know from when I used to temp a few years ago, every role I did was offered to me during a telephone conversation when I was chasing up the agency having not heard from them for some time. Sadly, you can't just wait for them to contact you.