SEEK A SEPARATION

Just because someone didn't work out in your department or under your leadership doesn't mean that she couldn't still be valuable to your organization. Many people who are difficult are actually fish out of water. For whatever reason, they wind up in a job that does not suit their skills. In this case, it makes sense to consider moving them, if possible. You can consult with human resources about other opportunities within your company that would be a better fit. One word of caution: be sure there is sufficient distance between her new department and yours.