It's not acceptable that she has been unable to progress with her task in four days, has refused your offer of help, and isn't telling you why she's stuck. She's not doing a good job in her role.
However, she's an unpaid intern who has never had a job before. She likely doesn't know any better. It is absolutely unacceptable that you have allowed someone who reports to you to remain stuck for four days on a task that should have taken a few hours, without forcing the issue. She doesn't need to tell you whether she needs help: you tell her that she needs help. You're not doing a good job in your role.
Fix your behaviour. Get involved in what she's doing and make sure she's doing it right. Discuss with her what she's researched so far. Explain some of the details verbally, engage her in a conversation about the subject in which she can ask questions, and let her go away and research from there. The sooner she gets through this research/reading that's required of her, the sooner she can do something interesting. She will probably benefit from having an outline of what the "real task" is that she's preparing for, even though before doing the research she maybe isn't equipped to really understand the task. Do these things right now, today, before she wastes any more time.
There's no need to blame her, just inform her what she needs to do in future. Likewise there's no need to blame yourself as long as you get on with it now that the issue has proved unsustainable.
This will take some of your time, but that's what taking on an intern means. Short-term workers who can get on without taking up your time aren't called "interns", they're called "contractors", and you pay them ;-)
The same goes for attitude: she probably doesn't know that as a junior employee in your organisation she is not permitted to be relaxed, and is not permitted to treat the rest of you as she treats her peers in class. In some organisations that would be permitted, at least to an extent, but even if it wasn't everybody needs to learn things somehow. You're supervising her, so you have to tell her the rules. If you think that telling her the rules is "giving her a hard time" then you might need to reflect on whether the rules are good rules. You might like to tell her, "look, this is going to sound really unfair, and everyone struggles through it, but in order to prove yourself in a junior role there are certain hoops you need to jump through". Then she knows she's being tested here, she's not hanging out among friends.
So, accept that you must present her with the facts, and what's expected of her, plainly, by the end of tomorrow. I don't know how you can best present them because I don't know your organisation, but I'm confident you'll figure it out pretty quickly once you decide that's what you have to do.
Praise her as soon as she improves. Recognising what she does right will go a long way toward avoiding making this a hard time for her, and it's also an effective teaching method since both positive and negative feedback are required to learn what is good and bad.
It's possible that she doesn't care about the role, doesn't care about what your company does, knows what she wants in a job and this isn't it, but her course has a forced labour component she can't avoid. There are still things you can teach her about how to be an employee, and maybe you can find something she would like to do, but if the situation sucks that badly from her POV before you even got involved then there's only so much you can do to mitigate it. However, I don't think you can fairly judge that based on her first task, so assume good faith on her part for now.