Don't beat yourself up. This is a difficult situation even for someone who is experienced with mental health issues. You can only do the best you can, you're human, afterall. (((hugs)))
The lack of desire to live is not the same thing as being suicidal. I know that sounds odd, but it's true. The lack of desire to live, the finding no joy in life, no motivation to get up and get out there and live it is depression. Suicidal is when that becomes so overwhelming they can't stand it anymore. I'm not so sure you're daughter is a real suicide danger at the moment. She has reached out for help, whether she wants to admit it or not, which shows that while lack of desire is there, there is still a part of her that wants to get better. That doesn't mean you shouldn't take her seriously or keep a careful eye on her, because you should. The line between the two areas is very fine and it's easy to cross.
I would sit her down to chat. Explain to her that no one will admit her unless she is an immediate danger to herself or others, unless she would like to be admitted for the evaluation and to get medications started under supervision. It's good that she went to the college counselor, again that shows she's trying, but I'm doubtful they're going to help her on the same level a private therapist or psychiatrist will do. While there will be no quick fix, I'd explain to her if she wants to feel better and function better she has to be willing to go into treatment and work at it. It's like if someone had abdominal pain and just sat at home willing it to get better, it gets worse, they continue to ignore it, they're in agony, then they may or may not be up to getting help at that point.....as that abdominal pain was appendicitis. I used to explain to Nichole that way, it was an easier way for her to grasp it.
Not all therapist or psychiatrist are equally as good either, so it's necessary to find a good "fit". That's not hard to understand, if you don't feel comfortable with someone, you're really not going to feel like opening up to them and being honest with what is going on. Which makes it really hard for them to help you.
If this is not behavior you've seen before, there might have been a situation that has caused this. If you can get her talking, if you have that type of relationship (I did with mine), then that might help and she might tell you what has brought this on. But that would depend on the cause, she might have valid reasons (in her mind) for not telling you too.
Such a hard place to be in.
((hugs))