It's hard to answer without all the details surrounding the circumstances of that ticket, and what happened at trial (i.e., default conviction for failure to appear, etc..)
Assuming that there was no criminal code components during that charge at the time of the ticket/conviction, I think you may have a good chance to have this appealed.
Afterall you are self-represented, and that is a valid excuse before the court for making some mistakes here and there. 2nd, there is a possibility to appeal after the limitation period, you would have to ask for an exception. I've heard of cases (serious criminal matters) where the consent of the Attorney General was sought several years later (in some cases 40+ years after the conviction) based on new evidence (for example: DNA evidence), so the possibility in law is there.
IMHO I think you should give it a try, nothing to lose, just file a leave to appeal motion with the court asking to appeal the case. You will be in front of a judge (or Justice depends on your court) and you will have to argue why should you be allowed to appeal the case this late.
Your grounds are pretty much what you described:
1. You are self-represented, have no experience didn't even know what steps to take or how long you have to appeal
2. This damages your entire career and you honestly feel you have a valid defence against the charge that should compel the higher court to quash the conviction if this appeal is allowed.
3. The personal circumstances, caring for a close relative that only had you to care for because others are unavailable, subsequently he/she died of cancer but at least you extended them the duty of care and courtesy that they deserve. (the word courtesy here is to send a message to the Judge that they should extend them the same courtesy by allowing this appeal)
4. Your personal Health condition: you were seriously injurged and unable to even carry out your regular day-to-day duties, let alone go to court, prepare a serious and strong legal defence, research case law and defend this charge properly.
5. Any other grounds you could think of.
Don't hesitate to back up your case with as much evidence as you could, if you coud present some medical records of yours, the relative, affidavit from your wife or anyone with you who could attest to your psychological/physical state at the time that didn't allow you to defend this (or defend it properly). If you had a lawyer/paralegal, you can always use the ground of ineffective use of counsel saying they didn't give you proper advice.
The Judge does have that jurisdiction to allow the appeal, that's why you file this motion in court. If you convince the judge that there are valid justifications for you not defending this properly before, and that the interest of justice demands that you are given the chance to appeal then the appeal may most likely be allowed.
If the judge isn't 100% convinced, he/she might delve into the details of your appeal (i.e., your grounds, error in law, fact, what was missing from the defence, etc..) so be prepared with that information if this happened.
You might research case laws where some leave to appeal after the limitation period were allowed (check canlii.com).
Finally, good luck and hope you can get back on your feet quickly.