Dear Neha Jain,
Allow me to quote Winston Churchill, "A lie gets halfway around the world before the truth has a chance to get its pants on." I wouldn't like to be lied to by a new joiner, so I won't ask you to lie and end up in some embarrassing situation. You can defend yourself and the cause or reason only when you are in agreement with the truth. When you lie, you tend to complicate the situation and get stuck.
Formulating a Cover Letter
Now, only because you have asked, I would suggest formulating a cover letter mentioning your previous rejection of the offered position. You can write something like, "Accept my heartfelt apologies for declining the previous offer for this same position. I would like to bring to your attention that I had some urgent personal issues to address and could not keep the company uninformed of your joining. I would gladly state that I am now at ease and more than willing to take on the responsibility of working with your firm. Although I understand the position may have been filled, I felt it was my professional duty to inform you of my availability for the job. I hope my application will be considered..."
You can add any additional details yourself.
After paraphrasing, I would recommend sticking to the truth.
Regards