UPDATE:
Thank you to all who gave input and shared their knowledge and experience, truly appreciated. I’m sharing this in case anyone comes across someone in a similar situation or with a similar question in the future.
I went to Taipei City Hall, they give free 20 minute legal consultations to the first ten people in line each day.
I explained my questions and concerns to the lawyer and showed him the contract.
The first thing he asked was do you have a work permit? When I replied with “no” his eyes got big and he stated that “the contract is no good you can’t work without a permit.” He asked if the school provided me any training, to which I replied no (contract signed Friday asked to start Monday).
His advice was to communicate with the school via registered letter if possible, but email works as well, that I will not work illegally without a work permit. The lawyer stated that working illegally in Taiwan is a very bad idea (clearly).
He didn’t address anything about working at a kindergarten being illegal and that didn’t seem to be the issue for him - rather the part about receiving cash payment (which I have not) and working prior to getting the work permit.
I asked if I still needed to give one months notice as outlined in the contract, he said no.
I also inquired about the penalty, he said that the school can sue me, but the chances of that are small and if they do so “I think they will not win.”
He gave me the information for the Legal Aid Foundation in the event I do need a lawyer and advised me to contact them as well if I had further questions.
I did not mention the name of the school or any individual. I am not trying to make waves for them nor myself. Major lesson learned during my first week in Taiwan. I can provide more updates with how the school responds if you guys want. I imagine they won’t be happy, and understandably so this will be a inconvenience for them to find a replacement, but I doubt this will come as a surprise.