Jess, does your HK license include motorcycle endorsement? If it does, then get yourself an International Driving Permit in HK and bring that together with your regular license.
If your license doesn’t cover motorbikes, then any IDP you got wouldn’t either, and you wouldn’t be entitled to ride a motorbike in Taiwan.
I will be completely honest and say that you might have a more enjoyable and relaxing visit to Taiwan if you used public transport. There are two reasons;
1 Traffic here is dense and general driver behaviour is unpredictable. You have to concentrate hard, even on country roads. If you’re looking for a laid-back cruise this may not be the place.
2 Most rental bikes here are 150cc or under. I think they’re likely to be scooters rather than motorcycles. There are some larger motorbikes available to rent, but this is very expensive, and in addition the rental shops for these bigger bikes may only rent to Taiwan residents.
There’s information about a big-bike rental shop in Taipei here;
[Real motorcycles for hire
and a thread started by somebody with a similar question to yours here;
[Rental of Motorcycles in Taiwan
Regarding places to go; this thread itself has some good information if you can wade through it. There are also a lot of other good ideas in the travel forum. One more thing; if you do go ahead with the motorbike trip idea, don’t be too ambitious in terms of distances for your itinerary. As I said, the riding is demanding here, and you’ll also want to allow yourslef plenty of time to stop and see things.