Vtyjkyuk

Don't worry about writing an apology letter, try and move forward onto other opportunities instead.

As a priority, you need to work with your doctor on your health/medication issues and get your health and well-being resolved. When you've done that, you'll have a better platform from which to look for work.

Unless you get your health sorted out, you're likely to run into this same issue again and again and again.

As @snow indicated, a letter of apology isn't absolutely necessary here. That said, it would be a nice gesture and may make you feel more at ease with yourself. If that's the case, such a letter could be very short and simple:

Dear Interviewer:

Thank you for meeting with me on the date to discuss the employment opportunity. Unfortunately, I could not complete the process. I apologize for that. It is my hope that you have good fortune in filling the position.

Sincerely,

Your name

You'll want to change the italicized parts appropriately. Also, tailoring this to better fit your own writing style and the particulars of the situation would be good.

The above is assuming you're not going to pursue this job. There is no guarantee that they would re-consider hiring you, but if you want to try, a modified approach would be in order. In such a case, it might look more like:

Dear Interviewer:

Thank you for meeting with me on the date to discuss the employment opportunity. Unfortunately, I experienced a panic attack and thus could not complete the process. I apologize for that.

If the position is still available, I am still interested. If you would be so kind as to re-consider hiring me, I would like to come in for another interview. Would you be available on (dates when you're available)?

Sincerely,

Your name

To be honest, the odds may not be very good for you to be re-considered, so please don't get your hopes up too much. Of course, if they do call you in, you'll need to be able to complete the process.

As @snow said, taking care of yourself is necessary here, as you'll need to be able to function at work once you get a job (this one or another), and - more importantly - for your own well-being.

My advice is to not send a letter to the company. Generally, people way overestimate the amount of thought that's given to an applicant that's not given the job. One way of looking at it is: if you went to a car dealership, and looked at 8 cars... how often do you think about the 3rd car you looked at and decided you didn't want? Probably not a lot - you've probably put it out of mind, and are more focused on the car you did buy, or are excited about the car you're about to look at.

Well, it's the same on the other side of the interviewing table. If, after the interview, we decide that the person isn't the right fit for the job... well... we generally don't give them a lot of thought from that point forward. Why would we? We still have to figure out who we're hiring.

Sending a "Thank you for your time" email is a nice touch, but it's not something I'd really even notice if it didn't come. I definitely wouldn't care all that much for a long email/note from an applicant apologizing - it's fine, I've forgotten about it, and I've got other things I need to focus on now.

So here's what I'd recommend: Go ahead and write the apology letter, but don't send it to the interviewer.

Writing the letter might help you get some closure on what happened, or explore how you were feeling during/after the interview, and help put the experience behind you. And, best yet, it might help you do better at the next interview.

Do not bother with the letter.

Concentrate on your health.

Get that fixed and take the medication

The other answers already cover the main point, forget about trying to fix this interview and look for another one.

The point I'd like to add is planning makes interviews go smoothly. You said you were 20 minutes late because of the long journey. Leave 30 minutes earlier than you need to so you're always on time. You could even do a dry run: travel from home to your interview location a day before to check how long it takes and make sure you won't get lost finding it.

I think this is important because you said you had a panic attack and I think the beginnings of that attack came from being late (of course it being your first interview in a year would also be stressful).

Before the next interview practice with a friend. They ask you 10 standard interview questions and you practice your responses. Another trick I do is research a company beforehand and memorise 3 key points about them. This really impresses interviewers.

i.e.

Interviewer: So what do you know about our company?

Me: Company X was founded in 20XX, you specialise in X,Y & Z and lead the market in X

Other answers are good but here's another option. If it would make you feel better, call the company instead of writing a letter. This would have the benefit of not leaving a paper trail encase you didn't want them to have on record you suffer from mental illness. Keep it short and simple and say

It's Rowan Mohamed calling. I would like to apologize for not
completing the interview last Thursday. I get panic attacks and had
one in the moment.

They may ask you if you want to reschedule so have an answer to the question before calling. To decline, you could ad

I've decided I must focus on my health at this time and not able to consider this position

Of course, like the other answers say, you could just do nothing.

What is far more important than writing a letter of apology is for you to understand that

1, the job hunting process is stressful

2, it's stressful for nearly everyone, whether they're recruiting for a job or looking for a job, so it's natural and normal for you to be stressed

3, for some people (such as yourself) it is more stressful than for others

4, This is the important bit you will only make it less stressful by a) preparing and b) pushing yourself through the pain, so that you get habituated to the process, and somewhat inured to the pain. There is no other way around it. It takes a certain amount of faith, but it is true.

Simply, you would have been better off if you had gone through with it, but you gave up. I understand why you did it, but you need to get past this. The interview was just one interview. Next time you go for an interview, just repeat to yourself: this is just one interview, and regardless of the outcome I will feel better afterwards for having gone through it than if I bail for whatever reason. And make sure you are prepared next time: about the position, the company, and crucially where the damned interview is; and make sure you get there 1/2 hour early. Maybe you can grab a coffee or something whilst you mooch around.

Separately, it sounds like you need some help with your mental health. We have an NHS for this. Maybe you can get some help. Speak to your GP. Good luck.