That patient was mentally ill and bought into a cult. She was irrational, out of control, and spiraling. Killing herself would have solved nothing. Most people who would have heard about her story would have assumed that she was mentally ill, and they would be right. That said, she would eventually kill herself anyway, so unless the doctors were prepared to find a way to have her institutionalized they were wasting their time.
IMO, there is no doubt that the people (cult leaders) that were running that website were in no way going to kill themselves. That's what that cult's followers are for.
The only way to try and get that woman to change her mind was to ask her why weren't the cult leaders killing themselves? But she was so irrational, she probably would have died before the doctors had any hope of changing her mind about killing herself.