Australian working holiday in Taiwan. What jobs are easiest to get?

There is no such thing as a "working holiday" in Taiwan. All foreign workers require a work visa. You'll need a sponsor to get this and it can only be obtained outside of Taiwan.

There are English schools and it is possible to get a job, but I don't know if they will take just anyone or if they require TEFL certificates.

I would think you would have known better, but saying that you speak "conversational Chinese" is meaningless. If you speak Mandarin or Taiwanese, that would be helpful in Taiwan. But "conversational Chinese" could also include Cantonense, Shanghaiese and other Chinese "dialects" (I hate using that term as it's not accurate in my opinion, but that's how the Chinese view it). In fact, in my country (USA) saying that you spoke Chinese for over 100 years very specifically meant that you spoke Cantonese as that and not Mandarin was the language that Chinese immigrants spoke when they came to our country.

Actually, I believe Taiwan has reciprocal working holiday visa agreements with Australia, Japan, New Zealand and (possibly) the United Kingdom.

Most schools require one year contracts. I've never heard of anyone teaching English on a working holiday. Maybe such programs exist.

Jman- It's clearly stated on the official government websites that Australians can get working holiday visas to Taiwan.