I gave you the short answer, but I can see you are curious about the long answer.
None of us are lawyers (I don't think), so please understand that all of this fabulous free advice is coming from amateurs who do not know all of the particulars of your situation, or all of the precedents in the Japanese labor law. It is easy for us to give you advice based on our experiences, but doing that could lead to disastrous results for you if there are some things about your employment history or conditions that we don't know.
Having thus cleared my throat...
Your employer has already demonstrated a disregard for the law by prohibiting you from taking a day off (among other things). She is violating the law while at the same time threatening you with legal action. It's no longer a polite conversation. It is now a knife fight.
You are correct about restrictions placed on contract employees, but you are permitted to terminate your contract for "unavoidable circumstances". The civil code does not explicitly state what unavoidable circumstances are, but I can say with confidence that your deteriorating health would be an unavoidable circumstance. And since health is mentioned in your contract as a valid reason for leaving, you have a bulletproof argument. The relevant civil code is 628.
有期雇用契約を破って退職したいのですが…
期間契約者の途中退職
http://www.sraf.info/index.php?民法第628条(やむを得ない事由による雇用の解除)
Regarding the period of time to notify your employer: civil law states 14 days, labor law states 30 days. Labor law takes precedence over civil law. However, labor contracts are covered by the labor contract law section of the civil code (627), which states the time required to notify varies depending upon whether you get paid an hourly wage, monthly wage, or annual wage. Check out the site here for more info. I assume you get paid a monthly wage, so 30 days should suffice.
http://www13.plala.or.jp/S-Kawamura/roudo/taisyoku.html < determination of time needed to notify termination of contract.
Regarding the potential of your employer suing you for leaving the contract early; as mentioned above, the law allows for you to leave your contractual obligation in the case of unavoidable circumstances. This does not safeguard you from being sued by your former employer. She has the right to sue for damages, but the majority of sites I have seen say that the likelihood of her winning such a suit is remote. The law doesn't like to see employers punishing their former employees. Plus, she would have a difficult time proving that you damaged her business. In any such suit, her labor practices would come to light, and I think that would be very inconvenient for her. But the fact remains, it is within her rights to sue you for damages, regardless of how weak her case seems to be. My guess is that she will not dare to do this, but I put the possibility here because many people mistakenly think that it is somehow forbidden for her to do this. The thing she is
not able to do is to put a pre-determined amount of damages in your labor contract. If she just says that she might sue you for damages if you leave your contract early, that is OK and doesn't violate any laws.
http://j-net21.smrj.go.jp/well/law/column/post_181.html
退職と手続きのルール | 労働基準法違反を許すな!労働者 < 会社側に損害賠償などを求める権利が発生する事もある
But, like I said, who cares about the law during a knife fight? I refer you back to my original post. Do you have the stamina or the resources to fight her in court? If yes, give thirty days notice, take as many days off as you have accrued, and get ready to fight. See the labor standards people or a labor lawyer as you see fit. Note that it has been my experience that both labor standards people and labor lawyers are nowhere nearly as aggressive as we would expect them to be. In fact, the overwhelming impulse seems to be to find an amicable situation, rather than to get aggressive. If you don't have the stamina or the resources to fight, work until the 20th.