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 wont like getting lied to be by a new joinee, so I wont ask you to lie and end up in some embarrassing arena. You can defend yourself and the cause/reason only when you are in truce with the reason. When you lie, you tend to animate the replies and get stuck up.
Now only because you have asked, I would say formulate a cover letter mentioning your previous denial for the offered post. You can write something like'' accept my heartfelt apologies for my denial of the previous offer for this same very post. I would like it to bring it to your notice that I had some urgent personal issues to address and could not let the firm in ignorance of your joining. I would gladly like to state that now I am at ease and all the more willing to take the responsibility of working with your firm. I understand that the position might be closed but I felt it to be my professional obligation to let you know about my availability for the job. I hope my candidature will be considered..."
You can add the knick-knacks by yourself.
Having done that devious paraphrasing, I would suggest sticking to truth.