You need to look at this kind of problem with both performance and behavior aspects:
Performance of the Person
Since you already said that people are satisfied with her performance as an employee and she is also very cooperative, then you cannot nail her on the performance issue.
Behavior
Every person has the right to live her life as she wants; however, if she is married and having an affair, this, as per our judicial system, is illegal and is called Bigamy. Now, I would not suggest writing any letter to her family or to any third person since you do not want to be a moral police for society. What you should focus on is cleaning the environment of your organization. Now, if you ever get any report that they are behaving indecently in the office premises (please remember that the office premises do not mean only the building but also the parking space and the cabs if provided by the office), found in compromising situations, or using foul or unacceptable language or gestures, then that is the time to give her the feedback, and the following steps can be taken. They can go in the order specified below, or you can also jump to the particular step based on the severity of the situation.
- Verbal feedback
- Warning letter
- Termination
However, please remember that while using the first two options, you would need to have a discussion with her. In that discussion, you need to communicate in very strong words that "I know that you and the <other person> share a very strong relationship. This is your personal matter, and I would not like to intervene; however, I need to make sure that the office ethics should never be forgotten. So, I would like you to ensure that whatever you have in between should not be visible in the office or office premises. You do whatever you want outside, but in this office, you should behave like normal colleagues. Also, ensure that this message is communicated to your partner too. If you are found guilty of anything like this, then we will have to take stern action."
Please remember we will not involve any third party in this case.
Regards,
Kanwaljit Singh