Okay, so I don't really care for Lily, she annoys the hell out of me.

I gave up and checked the wiki and it says to get her to play the tape of the grandkids I asked about Leo, seen a psychotic break, and slept. I've been wandering about for ages and haven't had anything pertaining to medicine happen, I talk to her rather frequently and no new options appear.

Is there something I can do to expedite this?

For me it happened after combat.

Thanks, I guess I will try fighting more. Maybe it's time to kill some Fiends again instead of questing.

Yeah, didn't come up for me until she got into a fight in which she fell below 25% health (which is directly related to the need for medicine, as you soon find out afterward).