You are at your wits end because of your Associate Director. Your take is that she lack qualities or substance that Associate Directors expected to possess. For this you would like to address a letter to the highest authority.
Well gentleman, when the lady was recruited, your Managing Director (MD) must have interviewed her and she must have satisfied MD. Going further, she has completed her two years in her current job. Therefore, MD must have been satisfied with her performance. Against this backdrop, it is better to drop the idea of sending letters to MD.
I admit that thought of writing letter to MD must have occurred to you because you perceive that she is causing losses to the company because of her unfitness or lack of leadership qualities. Whatever her capability is, she is occupying present designation with the imprimatur of your MD. Whatever her performance targets may be, MD feels that she is fulfilling those targets. Therefore, please ask question yourself on appropriateness of meddling between two seniors.
It would be a myth to believe that by writing letter, your MD will take action against Associate Director. You are employed in your company to deliver certain level of performance and howsoever loyal may you be, you are not there to reform someone. That is not your job but MD's. If he were to be that sensitive then he would have taken feedback from the personnel working in your department. Why has he not done that?
In this case I feel that there is something more than meets the eye. The lady's presence could be because of trade off with the gains that your MD is accruing. His benefits could be invisible to you. Else MD is forced to continue with her employment because of the pressure from unknown quarters.
In view of this, I recommend you to cool down and look for another job. That is most sensible solution. It is better to abandon thought of reclamation.
All the best!
From India, Bangalore