Dear {Name},
I am fully aware that I didn’t pull my weight on {joint project}. I am so sorry for contributing to the stress and workload of the rest of the team. I owe you all a debt of gratitude for covering for me when you could have rightfully thrown me under the bus.
I plan to let {boss/board} know that the rest of you had an outsized role in the success of the project. I will also let them know that this is a one-time failing on my part. Rest assured that this will not happen again.
I should have been more open with you. I was dealing with {personal matter/unusual issue/specific diagnosis}. I didn’t feel comfortable talking about it at the time, even while I knew I was basically phoning it in at work.
Again, I am so sorry for putting you in such an unenviable position. I hope that you will be open to working collaboratively with me in the future.
Sincerely,
{Name}
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Index of letter of apology templates