Write from facts rather than momentum
Pause long enough to remove accusations, assumptions, and details that belong to someone else. For a message about how to apologize for losing something you borrowed, use this factual base: name the exact action, the inconvenience it caused, and what you understand now.
Offer a proportionate next step
Match the size of the request to the relationship and to what the recipient can reasonably decide now. Offer the smallest useful repair and give a realistic time for completing it. Make the immediate next step clear enough that the recipient can answer without reconstructing the entire history.
Protect privacy on both sides
Do not inflate the apology, fish for reassurance, or demand immediate forgiveness. If the person has asked for no contact, repair what you can without sending another message. Use the most private legitimate channel available and do not share the exchange to recruit support or pressure.