I'm having some trouble figuring out the point you want to make, might be because you're not allowed to talk about the case, so I don't really see, what exactly the problem was. Did she expect psychiatric help from you and was offended when you told her that we aren't trained for that?
Did she just not accept the solutions you offered? Did you feel like she was being ungrateful? Did you think she was just looking for external factors to blame without considering her own involvement?
Did you feel like she just wanted to rant?

In other words: What was the reason, you felt you couldn't sucessfully continue providing help? And what was she trying to manipulate you into?

I concluded that you thought for yourself that in the end your negative feelings towards the client overshadowed your ability to help, if I'm not mistaken? That's one big reason why I think a team of professional social workers with different characteristics, skills and methods is crucial for good social work.
I also agree with the need for supervision to get the problems and questions you're confronted with in this complex work environment off your chest. It's too bad that often times it's just too expensive for the freelancers or employers.

Could you try clearing up my questions for me? I understand if it's not possible to go into details to protect the client.

Oh and I do think the exchange of private practice examples would be interesting, just to see what other people in the profession think.

I'm always quite amazed what kind of different content can be found on here. :)