Yep so essentially, being "pushy", be it for trying to do things or keeping saying that it's all fine is likely going to be very counter-productive.
As you've noted it's not her talking, but her depression. Trying this approach with someone who's got an independent mind is only going to increase her guilt and "fake" feeling that she'd rather you let her go. It is also very difficult for people like her to accept that they need help when they've built their whole life around a need for independence, especially from sentimental/familial causes.
My best advice to you would be to be extremely patient, as in give her some space, and if there's a hint of her wanting to do something, reel it in but be very cautious and gentle.
Also, meds WILL help her, but they're hell at first very often, so be wary of that. For such a person I would also think that a psychiatrist is NOT the right person to go. She should try a psychologist. Also, you mentioned academic failures, so if she's works at a university, there are often councelling services that could be a good first start. Although that might not be the case in every country, dunno where you live.