This should be an easy fix for you.
In your plan you need 3 elements
1: What went wrong
Do not deal with the specific family emergency, since it is not relevant.
What went wrong was that due to unforseen circumstances you were unable to ship orders on time, and you did not have a contingency plan in place to fall back on

2:What you have done to remedy the situation.
i’e Dispatched all late orders by expedited delivery and emailed the affected buyers to apologize for the delay.
Instigated a contingency plan in case of future issues
This might be to extend your handling times, so giving you longer to ship your orders
It might be that you have appointed a trusted family member or friend to take over the running of your account if you are ever unable to do so [whether that be because of an emergency or illness or simply a holiday]

3: What steps you will take moving forward to ensure continued compliance
This might mean monitoring your accounts ODR rate, changing your work practices to enable you to ship faster, or adding a second responsible person as a secondary user so that they can also access the account if you are unable to.

Keep your appeal concise and to the point, do not add any unnecessary waffle [they don’t need to know he details of the emergency, the name of a nominated person and nor do they need you to humble yourself in your appeal]

Good luck.