I think you can get a pretty decent stay with US$60-80 per day including accomodation, transportation, some petty cash for food/drinks. Some places can be over-priced, but if you look for reasonable, average areas and mingle with locals, you can get good deal enjoy cheap (and amazingly good) food island wide!
References :
I live in Taiwan!

It depends on the level of hotel you want to live, for example: a five-star hotel may cost you $150~200USD per day, if you wanna spend less you can travel around by bus instead of taking taxi or plane.
References :
my self.

I think US50-80 is enough(do not include the hotel fee). The mass transportation fee is inexpensive there. In average is about US 0.5-1. Taxi is also not expensive. You can also have excellent food in excellent traditional chinese food restaurant in about US10-20 a set meal. If you want to visit the country side, the bus and train is also convenient. The English might be some little problems there.
References :
I am a local tour guide in Taiwan.

$2-3/meal box so 3meals/day<$10

$5/day on transportation should be enough unless u travel to different cities everyday

$40-60/night on accommodation
References :
me