You might have a problem. I’ve never heard of anyone successfully filling in that blank on the birth certificate. The birth certificate is of dubious value anyway (even when it’s wanted) because it doesn’t show information that isn’t in the household registration documents, and foreign governments (well, the UK’s and Canada’s) always want those (you can see what the UK wants from you for your child’s passport in my post here: UK Child Passport Application (note FCO Birth Certificate for British Citizen born overseas is worthless))
If your wife’s household registration documents (show her the words 戶籍謄本 and she’ll know what this is) have you as the father, you’re probably in luck as far as getting a UK passport for your daughter. If they don’t, you’re going to have to handle it in the UK. It’ll probably be easier to make some sort of sworn statement of paternity there anyway unless you can only go the DNA testing route.
Sorry it’s not much, but you’re welcome. You can pay it forward by updating here with your experience; most people who come here with urgent questions like yours never come back to improve the forum for the next person with the same question.