Well, this is taking longer than you thought it would. What happened to a heartfelt, "I'm sorry." (?) Small offenses are salved by a heartfelt I'm sorry, and since I'm guessing you tried that and it didn't work, this might be a bigger offense than you thought. It might seem tedious to make sure you cover these 6 bases, but it's definitely better than having several marathon conversations/fights where you go in circles, at least one person cries, everyone's exhausted, there are tons of passive aggressive vibes all over the place and nothing is resolved.
The good news is that steps 4 and 5 are probably the easiest steps involved in giving a meaningful apology. After you've done them, there's only one step left. It's the one thing to never, ever do while apologizing. (But, we'll get to that in Part 5.)
You've done all the thinking involved in Part 1 and you get that apologizing is a process, not an event (just like forgiveness). You've also used this handy list in Part 3 to help you identify just why exactly the person you hurt was so hurt.
Now it's time to take that hurt and make it feel better.
Step 4: Identify ways to actively salve the main hurt in the other person.
Verbally reassure whomever you're apologizing to that they're important, then show them that they are. Don’t say, “I’ll do anything to make this up to you.” That’s just well-intentioned laziness. It’s not the other person’s job to come up with ways to make it better, it’s yours. Lets say you really hurt your sister and she feels like she was left behind. You can say something like, "I know you probably felt left behind, and that's the last thing I want you to feel. You matter so much to me and spending time with you is important, it truly is. I'd love to have you over for dinner this Friday, and I've cleared (X night a few weeks later) to do (Y thing you two love to do together) if you're not busy?"
Step 5: Be reassuring about the future.
It's also extremely helpful to specifically say what you'll do in the future to try to avoid a similar misunderstanding. Show off your self-awareness and reassure the person listening to your apology that you get it now, and you're willing to take extra steps in the future to try to curb this type of hurt.
You're on a roll. Now you just have to dodge the last pitfall in a well-intended apology, the one thing you should never, ever do while apologizing.