sincerity

[ziawj2]

I lost a book borrowed from my friend. And I wrote to her. At the end of letter, I wrote:
1)I promise to buy a new one and give it to you in three days.
2)I know it is valuable so I will do anything to compensate you. Hope you wouldn’t mind.
3)I’m really sorry about that again and hope you can understand.

I think the second ending is better in terms of sincerity. What is your opinion? Do you have any other advice?
And In your apology letters, do you end the letters by writing "Hope you wouldn't mind" or "Hope you can understand"? Are they acceptable to native speakers? Thank you!

[emsr2d2]

I lost a book which I had borrowed from my friend so I wrote to her. At the end of letter, I wrote:

1)I promise to buy a new one and give it to you in three days.
2)I know it is valuable so I will do anything to compensate you. I hope you wouldn’t don't mind.

I think the second ending is better in terms of sincerity. What is your opinion? Do you have any other advice? Thank you!

Surely your friend knows whether you are a sincere person, so she will know whether you really plan to buy her a replacement book.

With your second example, do you hope that your friend doesn't mind that you lost the book or that you will do anything to compensate her for the loss. If she is a good friend, your second sentence is rather formal.

I would write something like "I'm really sorry. I've lost that book I borrowed from you. I know it was valuable but I'll buy a new one and get it to you as soon as possible. I hope that's OK."