When you say you have a work permit, do you mean you will have a work permit?
If by “visa for internship purposes” they mean a working holiday visa, then the visa is your work permit, and there’s no separate application.
Aussies have a six month limit on employment (per employer) when using a WHV, and iirc you also need to apply for an extension or fly out and in after 180 days (so a few days before the employment time limit if you start working on day 0).
You also have iirc a 3 month(?) limit on studying. BOCA’s website should have the details.
http://www.boca.gov.tw
This thread also clears up a few things: