Dear Kumar,
You are at your wit's end because of your Associate Director. Your take is that she lacks the qualities or substance that Associate Directors are 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 the MD. Going further, she has completed two years in her current job. Therefore, the MD must have been satisfied with her performance. Against this backdrop, it is better to drop the idea of sending letters to the MD.
I admit that the thought of writing a letter to the MD must have occurred to you because you perceive that she is causing losses to the company due to her unfitness or lack of leadership qualities. Whatever her capability is, she is occupying the present designation with the imprimatur of your MD. Whatever her performance targets may be, the MD feels that she is fulfilling those targets. Therefore, please ask yourself about the appropriateness of meddling between two seniors.
It would be a myth to believe that by writing a letter, your MD will take action against the Associate Director. You are employed in your company to deliver a certain level of performance, and however loyal you may be, you are not there to reform someone. That is not your job but the MD's. If he were that sensitive, 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 a trade-off with the gains that your MD is accruing. His benefits could be invisible to you. Else, the MD is forced to continue with her employment because of pressure from unknown quarters.
In view of this, I recommend you to cool down and look for another job. That is the most sensible solution. It is better to abandon the thought of reclamation.
All the best!
Dinesh Divekar