Dear Sir/Madam,
I am writing to you to offer my sincere apologies regarding not answering to your letter, which you sent me some time ago and in which you asked me to return the book. I know how much this must have upset you, that I have not returned this book so far, but I hope that we can resolve this matter agreeably.
This was due to the fact that I was convinced that I had returned this book to your library because I was in it a few weeks before I received your letter. I would like to say how sorry I am because yesterday I found this book quite by accident when I looked through my bookcase.
Please let me know how much it cost that I have kept this book for such a long time and I will gladly pay for it. I am sure that similar matter will not have place in the future.
Once again, please accept my sincerest apologies for the inconvenience.
Yours faithfully