Do I include the job I had for 3 weeks? I'm not really keen on explaining to an interviewer that it didn't last long because I had a mental breakdown in an office.
You generally want to be honest on the CV/resume about past job, especially if they immediately precede the job for which you are applying or are within the period specified on employment application. There are legal reasons for this, namely that failure to disclose a prior job on the resume is grounds for immediate dismissal by the company if they ever find out, e.g. based on a background check or if you ever mention it by accident and HR finds out. If you are dismissed for failure to honestly portray past work experience, it can cause issues down the line as well. Best to tell it like it is.
Note: In some cases there may be valid reasons to omit past jobs that did not last very long and/or were unrelated to the type of job you are applying for. Examples are hourly low-skilled labor types of positions you held during college, which can be safely omitted after you've held one or two professional jobs since graduation, because the time during which you held odd jobs is covered by the period when you were a student (which is covered under the Education section on resume).
Typically job duration is stated in month units so you could put "July 2016" or July-August 2016" (if the 3 weeks spanned more than one month). List your tasks/responsibilities during the time in that job as you would for any other job, but obviously keep it short and to the point since there wasn't time to get much done. One or two bullet points should do.
If/when they ask you in the interview why you left that job, say it was "for health reasons," or (longer version) "due to a health issue that occurred after I had started that job." They should not ask for any further details, and you do not have any obligation to disclose any such details.
The larger issue here that you may want to think carefully about is, are you at a point where you are ready for a new job? The reason I say this is it would not be good to have the scenario repeated with yet another job which lasts a few weeks or a few months.
Having seen someone I knew struggle with keeping it together and having to leave a job as a result, I would advise to invest in proper treatment and not to rush returning to work, unless returning to work is better for your mental health than not (i.e. for having that structure and distraction that a regular job provides). Good luck!