Back in 2000 I wanted to become a Resident Adviser in college. We had to read the book 7 Habits of Highly Effective People which contains very thorough ideas on how to apologize These ideas stuck with me and have helped me patch things up after my many many many many mistakes.
I think it goes something like this(totally paraphrasing from memory):
Explain to the person the mistake you made
Explain how you are going to try to make sure it never happens again.
Explain that you are truly sorry for it.
Ask the person to try to find it in their heart to forgive you.
I definitely need to to that right now. I stepped into a new situation that was easy to make mistakes in. I made them. I at least have the sense of calm that this tool can help me structure the apologies i need to make.
Extra note:
In apologies I think ego is your enemy(Holliday). When trying to reconcile with another person I don’t think ego can help you much so long as your make sure you aren’t destroying yourself by going too far. In fact taking a moment to see where your ego(not in a freudian sense) is pushing might be a great compass. Try heading in the opposite direction from where the ego, that compass is pointing you.
Here's an intense(kinda scary) inspiration of these ideas