Dear Richa,
This is a common issue that the HR professionals face in their profession. However, here, since the issue is with your report, it's relatively easier to handle. Why can't you speak to the person with all the leaves details she has availed? Sometimes, the person may be having very genuine constraints to do so. Understand the real reason for this indifferent behavior (availing too much leave) and handle it with empathy. Suddenly concluding that the MD has some other relationship with the person seems to be highly immature. This may be true or sometimes may not be true.
Tell the report about your expectations from that role and ask her to improve. Even if there are genuine reasons for her taking so much leaves, make her understand that she is playing an important role and such breaks are affecting the performance of the function. Tell her either to reduce the number leaves being taken and contribute on a continuous basis or you will have no option but to give the role to a more consistent employee.
From your mail, it is clear that your MD has categorically clarified that she has to report to you. So, I see a problem here in your inability to take control of the situation. If things are running fine even after your report's being on leave for so many days, the question arises is "is that role required"? If the absence is impacting the performance of the function, you may appraise your MD about the reality. I think he will agree with you that a function cannot be run smoothly with such employees who remain on leave every alternate day. Because, logically, he should be in a better position to understand the problem. So, your first priority should be to understand the views of your report as well as the MD on this issue.
If you present this case properly, I am 100% sure that your MD will agree for a replacement. Major part of this exercise should be done verbally rather than writing mails / letters. Once a verbal consensus / decision is arrived, that may be the time to engage on written communications. If I would have been in your place, I may have handled the situation as above.
MK Saseedharan