Thank you for sharing with us. I'm not a parent - I'm 18 myself, but I hope I can help a little bit none the less.
This is a very complex situation, and it must be very difficult for you to be stuck in the middle. I'm assuming she's entirely dependent on you financially, for transport, etc. I think it's important for you to know that as much as we want to change people for the better, especially as a parent, it's realistically up to your daughter.
I experienced severe social anxiety throughout high school and wanted to avoid going to school all together, but this was never an option for me because my parents wouldn't allow it. As much as I resented them for it at the time, ultimately it was the best thing they could've done for me, it helped me build much needed resilience. They were very much strict with me in terms of what I could and couldn't avoid, but at the end of the day, they were sending me to therapy sessions and were a great support system for me. I suppose it's what they call tough love.
Unfortunately like you said, I think without a push, your daughter will easily and quite happily stay in her room for the rest of her life. That's the sad reality about mental illness. Right now you are being a great support system for her in encouraging her to take care of herself and encouraging her to get help, she is very fortunate to have that. As your daughter, she does have the responsibility to transition into adulthood and gain her independence, and this isn't easy for everyone especially with mental illness.
I think doing things like checking in on her regularly while she's in her room and asking her what she's up to would be helpful to see what she's doing. Ask her whether she's done the things she's needed to and asking her why not. Hold her accountable for procrastinating. But always let her know that she can talk to you and ask for help whenever she needs it. Get her to show you how many TAFE units she's completed or what she's done for the day. If she hasn't been productive, ask her what she'll do tomorrow and that you'll check what she's done.
I hope you're taking care of yourself. Always remind yourself that with things like this their isn't always a right or wrong answer, and as much as we want to, we can't completely change how our loved ones think/act. But indeed we can help them out as much as we can and be a support system, and that won't be taken for granted in the long run.