Quitting Kindergarten..?

I have a friend that is part of the administration at a kindergarten and she really wanted my help to teach a specific subject to the babies and toddlers. We talked about the transition from one teacher to the other and I had a conversation with the principal about salary and hours (in my head it was a salary discussion, not wages, because we don't do that much in my country).

The problem is that, in my country, as soon as you start working somewhere, the company has to register your employment status with them. It's illegal not to do so, except if you're working there as a service provider.

Well, I've been working there since April, I talked to the principal about the employment registration process, he said he looked into it and was waiting for the paperwork from the accounting firm. I kept pressing, but we're in September and so far, I'm still not registered as an employee.

What makes this difficult for me is that I really like working with the babies and toddlers, the environment is nice and I like my coworkers. I'm also currently doing a post-grad in Educational Psychology, and being able to work closely with kids that have learning difficulties before I finish my studies is great.

No one mentioned I wouldn't be registered in any part of the process and since it's illegal, I didn't even ask.

The problem with registration brought other issues: I had to go to the hospital once for an emergency and I gave them my doctor's note. They should have payed me for the day, but since I'm not a registered employee, they didn't. I had to get surgery and I'm home still recovering (2 weeks), can barely speak, and they are also not going to pay for that. If I were a registered employee, according to the labor laws in my country, they had to pay. Another problem is that I had to work during summer break but the hours were reduced because they didn't have many students signed up for the summer camp activities, and my pay was reduced accordingly while teachers that are registered stayed home and received in full (again, I was not uniformed about any of this until I received less, and I wasn't even asked if I would be able to work during summer break until the week before it started, if I had a trip planned, my bad).

Additional information: Kindergarten is 17% of my monthly income, one 7-hour shift/week.