As a patient I understand and accept that the health care system is run by humans and that therefore mistakes can happen. I've had some happen to me in my care (nothing too drastic though). Your mistake comes under that category by the sounds of it. No lasting harm done which is lucky.
However I have had a relative whose very serious diagnosis was completely missed. They died some years prematurely as a result of inappropriate treatment exaggerating their symptoms. With hindsight it was very clear what had gone wrong and it was all down to human error.
Our family did not call lawyers or write blistering letters to individuals caring for our relative. Not because we weren't hurt or angry at the loss but because it was clearly a tragic mistake by professionals who were doing their best (judging by the other care recieved and their caring attitude leading up to her death) and we did not want to deprive the system of money and people it desperately needs. nothing any of us did by then would have saved our relative and that lawsuit/blame could have crushed the health care workers involved and maybe deprived other patients of a caring team. Some of us in the family had a face to face meeting with the team and that was it. We faced our loss, which is still sorely felt years later, and they will have learnt from it.
Some people will need to write a letter when a mistake is made in their care which is an understandable reaction. They may feel there wasn't the 'care' or attention they expected on the day or that generally 'the system' let them down. These may or may not be objectively true.
You are right to own your error and to personally reply and apologise to the patient which is hopefully all they will want- an acknowledgement that their problem has been heard by the person responsible (for the incident or for that part of the system). They will want to hear that you understand that it has caused them an avoidable problem/pain or whatever disadvantage they are complaining of, and that (IME a really important part) now they have raised this, their letter means that someone else in future is less likely to have the same problem happen to them. I.e. Their complaint has been worth something. You will respond by doing x or y differently in future as far as you are able to.
But please as everyone says, don't give up your job or beat yourself up or doubt your capabilities because of this. A lot of patients feel that the NHS is staffed by incredibly dedicated people, who are constantly being asked to take on increasing burdens and to continue to give us a caring service at the same time.
We really appreciate the (usually appropriate and often excellent) safety net you give us and the care we get and are very grateful for that. At the same time things can and do go wrong, but as long as the mistakes are acknowledged and learned from, there isn't more we can ask of any system run by human beings.