I am so sorry that I didn't get the package in the mail in time for {deadline/delivery target/etc.}.
I wanted to let you know that the {contents} are now on their way via {next-day mail/courier/etc.}. The package should arrive by {date}.
I apologize again for the inconvenience. {Explanation}. I'm taking steps to make sure nothing like this happens in the future.
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Index of letter of apology templates