I’m assuming you have a normal work permit (with an ARC) and work at a buxiban that has an attached kindergarten. If that’s the case, you should refuse to set foot in the kindergarten, and you should document any attempt to order you to work there. (For a long explanation of the legal issues involved in kindergarten classes, click here: Work Rules For English Teachers - #182 by yyy)
If they merely suggest, “Oh btw, we heard about an opportunity for extra work at some other school… What’s that? It’s a kindergarten? Oh, we didn’t know! Sorry, we would never ask you to work illegally!” or can plausibly say that’s what they told you, you don’t have a case against them on this point. At the other extreme, if they declare in writing that you’re going to teach kindergarten classes whether you like it or not, you have a solid case against them.
If you go to law.moj.gov.tw and laws.mol.gov.tw and read through these laws and regulations you can probably find other violations, e.g. holidays, overtime, sick leave, etc.
Labor Standards Act
Enforcement Rules of the Labor Standards Act
Regulations of Leave-Taking of Workers
僱用部分時間工作勞工應行注意事項 (part-time issues)
Act of Gender Equality in Employment
Civil Code (Art. 247-1, 483, and 487 in particular)
Employment Service Act
Check Art. 14 of the LSA for the list of situations in which you can quit without advance notice and receive severance pay within 30 days. (Severance pay is based on Art. 17 with reference to Art. 2 and to Art. 10 of the Enforcement Rules, but if you have a Taiwanese spouse the Labor Pension Act overrides it, basically meaning you get 50% less.) Any violation likely to affect your rights and interests is a legitimate reason to quit, subject to a 30 day rule.
Of course they are likely to resist a demand for severance pay, but if you’re willing to fight and able to prove your salary over the past six months and that you terminated the contract in accordance with Art. 14, you will win eventually. If they owe you holiday pay etc., you can also fight them for that, plus 5% annual interest on all late payments.
To use Art. 14, prepare a resignation letter in which you explain your reason(s) for leaving, choose the most suitable day (noting the 30 day rule), and email the letter to your boss. Of course you can resign in person, but you need proof, in case your boss pretends you just disappeared (which could get you in trouble later on). Another way is to give your boss a 離職證明書 form to sign, which you can apparently submit to the labor department for record keeping.
If you can’t prove any legitimate reason to quit, they still don’t have the right to deduct a breach penalty from your salary, thanks to Art. 26. It’s not realistic for them to sue over a $15,000 question, because a lawyer would probably charge more than that, and even if they have the brains to do it without a lawyer (and don’t owe you money for any violations), they would probably not consider it worth their precious time. 
The labor department in the city/county where you work can provide assistance.
If one of the reasons out of your control is the death of a family member, pay particular attention to Art. 3 of the leave regulations and Art. 62 of the Labor Insurance Act (with reference to Art. 19).