I wonder if you're a little freaked out because you can see yourself being like sister to this patient.. Going back to the Gloria videos:
Gloria: Yeah, and you know what else I was just thinking? I – I feel dumb saying it uh - that all of a sudden while I was talking to you I thought, "Gee, how nice I can talk to you and I want you to approve of me and I respect you, but I miss that my father couldn't talk to me like you are." (Touches her chin.) I mean, I'd like to say, "Gee, I'd like you for my father." (T: Mhm, Mhm, Mhm) I don't even know why that came to me. (Smiles)
Rogers: You look to me like a pretty nice daughter. (Pause) (C: Looks down.) But you really do miss the fact that you- you couldn't be open with your own dad.
Later Rogers would say: I was genuinely moved, I probably showed it, by the fact that she told me near the end of the contact that, uh, she saw me as the father she would like to have. My reply was also a thoroughly spontaneous one that she seemed to me like a pretty nice daughter. I guess I feel that we're only playing with the real world of relationships when we talk about such an experience in terms of transference and counter-transference. I feel quite deeply about that. I want to say, yes, we can put this experience into some such highly intellectualized framework, but when we do that it completely misses the point of the very immediate "I-Thou" quality of the relationship at such moments.
Is that a sin? Maybe part of you could see yourself being a good sister to her. Anyway, maybe it's time to discuss this dependence, say something to refocus on her life: "ya know, we can't work together forever, and I am here as long as you need me, but the goal is for you to not need me." Maybe discuss the yearning for the sister relationship and behaviors she can do to create more of that support.
Just dont start meeting outside of therapy, engage in physical contact, or text the patient. Don't respond to emails or spend more than a few minutes on the phone, you've got other stuff to do, set physical and time boundaries, but up frequency if needed.