It won’t help Libragirl right now, but, in case she decides to stay in Taiwan longer, she can get her Taiwan ID card. The question is not that she wasn’t born in Taiwan, the question is whether or not her parents old family registration documents still exist. My bet is that they do (Taiwanese local governments have very complete and extensive records).
So, what Libragirl can do is when she’s in Taiwan, she can go to the town/city of her family’s last known residence, and try to obtain a real certified copy of it. It would help if her father retained an old copy that she can bring to Taiwan. Then, she can go through the process to certify that she is the daughter of the father named on that family record. This kick starts getting the Taiwan ID process.
I know a South African woman who was in a similar situation and it took her about 1 year to receive both her Taiwan ID and ROC passport.
I also personally know of a man, not born in Taiwan, never lived or visited Taiwan prior, who was able to get his Taiwan ID card and passport in 1 month.
So, remember this, Taiwan is a land where the rules aren’t what they seem even if written on a piece of paper. You can get a lot of things done by speaking to the right civil servants and praying for a bit of luck.
I know, this doesn’t answer your question.