Dear {Name},
I am so sorry that I did not follow through with my commitment to {action}. I hate to make excuses, but just for context, {unexpected and extreme situation}.
It's completely understandable if this makes you less likely to trust me with {task/role} in the future. But I do want you to know that, if given the opportunity, I would not drop the ball again. I have now {resolved aforementioned issue}.
Thank you again for the opportunity to be a part of your {organization/goals/events/etc.}. I hope that we can collaborate in the future, and that I can have a chance to show you how I perform when I'm at my best.
Sincerely,
{Sender}
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