How do I handle this bad work dynamic?

I understand the old dynamics that employees will talk amongst themselves, but what about when my assistant vents to another support staff employee and consequently stops communicating to me. All I know is she has something stuck up her backside but when I ask it’s, “no, nothing’s wrong”.

I have done some reading on triangle dynamics but still have some details to sort out. Where does the blame go?
Where do the realms of personal and business start and end?

With regard to personal vs business, this is what I came up with:
When communications are stifled
When anyone is alienated
When morale is hurt
When productivity is decreased
When the companies culture is compromised.

All of these are happening in small subtle ways.

A little more; my assistant is fantastic outside of this situation. I realize she bears a lot of blame though.

The support staff person has told me she is all but leaving as soon as she gets a job with the hospital (she is studying to be a nurse).

This is a career opportunity to my personal assistant as she is in training to take over move as my office manager with a company performance split. I am very concerned that the other support staff person is undermining my assistant’s attitude about her job.
I am dealing with a lot of denial only later to learned my instincts were on the mark from same dynamics in the recent past.

What should be my course of action and when can I play it out.
I thought about giving them both hours where they were not in the office at the same time. Please don’t suggest just firing the support staff person if you don’t understand the complexities of firing someone. If you do understand and that is what I should do, do I have enough to let her go. What do I accuse her of when it is all subversive?