Although the difficult employee has now been terminated and is off the premises, your work is not quite finished. You need to announce to the rest of your people that the employee is gone. Call a special meeting, perhaps the following morning, if a staff meeting is not already scheduled. If anyone has issues with or wants to discuss the matter, make yourself available privately for a limited amount time, perhaps the balance of that week, to discuss it. Let them know that your decision to terminate or reassign the difficult person was, in many ways, a show of support for the dedicated hard work of the rest of your staff. You did it for their welfare as well as the overall best interests of the organization.
Termination is easier if you know you've done everything you can to save the relationship. Still, you may feel guilty about terminating the person, even someone who has made our life miserable. Yet, there are good reasons to follow the course of action you've chosen. Don't deny that your emotions are involved when dealing with difficult people. But don't allow your emotions to get in the way or lead you toward decisions and actions you will regret.