Well... here's where it gets tricky, yes?
You can't burden yourself with her decision to get treatment or not. If you were a baker and a client came in asking for cherry pie - you could offer her a cherry pie, or suggest she really strongly consider the apple pie because you know it would be better for her. But your client has the offer to take either pie, a chocolate cake, or no pies whatsoever.
I realize that in the field of mental health, we are involved with human suffering - and we are helpful people who want the best for our clients. Unless your client is hurting herself or others - or actively violating the rights of others - or is incompetent - you have no recourse but to offer your services and hope she will accept.
I know that this is TERRIFICALLY frustrating - but that's part of what we do... recognize a problem, offer a solution, and then our clients to make their own decisions about how to proceed.
Now, one option you might consider is to say something along the lines of, "I appreciate that you have some very important concerns... but really the only way to address them is with a face-to-face professional. If you were experiencing chest pains, you wouldn't seek help online - you would go to an emergency room to determine if it was a heart attack, appendicitis, or a bad roast beef sandwich, right? Well, the only way to better understand your important concerns is through a face-to-face meeting with a professional..." and then proceed to give her a list of local providers and/or hospitals.
Then... you have to let her make her own choices.
I'm sorry I can't be of more help... or that you can't with your particular client! It's often the nature of the beast, yes?
Best of luck to you - and thanks for writing!