This is a detour from "4 Steps To Be An Effective Engineer".
Let me shed some light on a simple strategy, which will help you deal with your mistakes in a responsible and efficient way.
First and foremost, you should understand this fact:
There is no experience without failure.
Give it some time to settle in your head.
Now the next fact:
It is not a real problem if you have made a mistake. It is a problem if you have not fixed the consequences.
Are you ok with that idea?
Then, here is the third one:
It is you who is in charge for fixing negative effects you have caused.
This is actually what distinguish a responsible person - she can deal with unforeseen events and respond to them in a sane way.
So what should I do?
Obviously, you should fix the problem.
Also, you should try to keep all the "involved parties" happy.
And here is a general overview of the things you need to do:
Do not wait for a command to fix everything. Fix it! Be proactive, but let everybody know that there is a problem.
Inform all your involved colleagues that your are dealing with the problem. This will add some certainty and calm people down.
Inform the affected users that there is a problem and you are solving it. This helps to build good relationships with end users.
Do not be afraid to show the problem - people love openness and they tend to forgive soon.
Ask others for help of guidance if you are in doubt! Do not make things worse if you are not sure what to do.
Escalate the problem, if you cannot make a decision! If you manager will bark at you - then he is just a motherfucker (or simply had a bad mood). You will deal with that later.
The foremost target is to fix the problem, so that people do not suffer.
Analyze the cause of the mistake. Do you understand how does your system work?
Did you have all the necessary knowledge?
How to prevent the same issue from happening again?
Give it some time.
You should really understand the strategy described above to be able to execute it in case of emergency.
Now, if you are not yet tired, I would suggest you to proceed with the article, explaining how to be efficient.