I've beaten this drum here before[1], but Dialectical Behavior Therapy, an evidence-based therapy program, has a framework for making requests of people with the acronym DEAR that can also be used for apologies. I think a good DEAR-apology would hit most of the techniques in this article:
* Describe the situation: "Accept responsibility".
"I didn't do X that you asked me to do, even though I knew I should have."
* Express feelings: "Express regret"
"I feel bad about it because I promised I would, and I had time to do it, I just didn't, and I don't have an excuse."
* Ask: "Ask for forgiveness"
"I'm sorry. Will you forgive me?"
* Reinforce: This could be restitution or repentance. I'd argue it's situational depending on whether it's a recurring thing (I'll do it in the future!) or in the past (I didn't defrost the chicken and ruined your dinner party, so I'll cook for the next one!)
Anyways, DBT has a few gems[2], and DEAR is one of them. I use it all the time for both asking people for things and apologies.
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[1] https://news.ycombinator.com/item?id=31859266
[2] Another one: panic attack or extremely anxious? Get a bowl of cold water and stick your head in it for around 30 seconds. Make sure your whole face, especially your cheeks, are submerged. You can use ice water or just cool water. Take a breath, and repeat until calm. It's based on the dive reflex and is a biological function, it will always work.