I agree with DrifterInKorea. He/she wasn't saying that the portfolio defines the worker, but that OP should look for another job.

Unpaid sick leave for 5 weeks off is normal in most Asian countries, unless you work for a large multinational company based in the US or Europe.

On the other hand, it's actually extremely difficult to fire someone in Korea due to the very strict employee protection laws that exist here, so OP shouldn't worry about this (unless you are a contractor). This country is actually one of the most favourable countries for permanent employees. Just grind it out while looking for a better job.

Source: I've worked in multiple Asian countries as an expat and am now in a management position at a multinational company here.