Have you tried talking to her about it first?

If I cared at all about somebody (and I care about most people more than to want my first step to be reporting them and causing them to lose their jobs), I'd take a moment to ask them if they've thought it through. I'd start with "What's going to happen when someone finds out? Because you know someone will find out." and try to lead her to the conclusion that she should resign and find another job instead of being fired when somebody eventually sees them together.

I'd also question how ethically skewed it is. How much opportunity for abuse of power is there? Does she directly handle his patient care? Has she taken steps to ensure she won't ever have to? Honestly I know more than one person who has met someone while working whether it be the patient, family member, etc. It works a lot better for us because we're inpatient hospital floor workers so they're with us for a few days and then gone, so there's not continuity there.

The biggest thing you need to tell her is to shut her fucking mouth about it. She doesn't need to be telling anyone.