Dear Lily,
The purpose of my writing this letter is to express my sincere apology to you. I broke the cup on your desk yesterday. I know that's something very important to you, so the following is an apology.
This is how it happened. As I was still studying in my dormitory yesterday, all my roommates went out. I want to do a big cleaning of the dormitory when there are very few people in the dormitory. Because our bedroom is too dirty, a lot of black handprints on the wall, the table is also very messy, there are a lot of waste paper on the ground. I had finished sweeping the floor when I was about to mop it，I accidentally touched the cup on your desk with a mop. The cup fell straight off the table. Your cup hadn't broken yet, but it fell apart when I picked it up. I express my sincere apologies for my carelessness. Please be sure to forgive me. If there's anything I can do to make it up to you, please let me know. One of the compensation plans I came up with was to pay you for a cup that was almost similar. I thought you once said that cup was a limited edition bought online. Please tell me the website of that product. If it is sold out, I will go to other ways to buy from others. I'll make it up to you for the exact same one, and if I can't I'll make it up to you for a very similar one. Do you think that's okay? If you don't think it's gonna work, you can ask me.
Once again, I sincerely apologize. I hope you must forgive me. Looking forward to your early reply.