One of the problems of living in Taiwan is the lack of a consulate for most countries, so for long time foreign residents of Taiwan whose passport expires, most countries handle renewal of passports in nearby countries such as Thailand or HongKong where full embassy services are available.
I had to do that a few years when my passport expired, and I’ll have to do it again in a few years (my! how the time passes). Fortunately most Western countries hold some kind of diplomatic relationship with Taiwan so they have representative offices that can help you handle these kinds of affairs. There are a few places where you can find out this information such as here.