Most of this kind of things are explained on relevant authorities’ sites, unless your case is extremely complicated. and when your case is extremely complicated, the possibility of people know how to do may be very low. so it may be better to ask to relevant authorities directly. In your case, relevant authorities are TECO in Philippines on your passport, and Household registration office in Taiwan on acknowledgement or adoptation by your father. HHR office does nothing on your passport, other than issuing some documents for your passport application.
You may already check, but I put a link to TECO’s page explaining how to get a passport. Required documents and how to prepare them are explained.
What are the requirements and procedure in applying ROC Passport（First Timer)?
https://www.roc-taiwan.org/ph_en/post/81.html
Notices:
3. When the applicant is illegitimate:2) If the father is an R.O.C. (Taiwan) national while the mother is a foreigner, the father must first accomplish an acknowledgement record (if the father has household registration in Taiwan, he will need to submit the birth certificate and the mother’s certificate of no marriage-CENOMAR and other relevant documents to the household registration office in Taiwan to register the Regarding the details of this matter, please consult the household registration office where the father has household registration), only then the child may qualify to possess R.O.C. (Taiwan) nationality and apply for the passport.
The name on your cirth certificate is the same with his English name on his passport? His date of birth, address, etc. are recorded on your birth certificate? If so, I think it may be OK. If it is not so, or even it is so, Household registration office might request additional documents to prove your father daughter relationship, such as his affidavit, photos of your mother and father and you, old letters from/to him to/from your mother, DNA test, etc. You father should ask at Household Registration Office what they need to approve his appplication to acknowledge you.
If you cannot prepare your mother’s certificate of no marriage-CENOMAR for the time when she pregnant and gave birth you, you maybe need a court order to declear that you are his daughter by DNA test.
This post and the thread might add some info.
After living in Taiwan for a year as his daughter on your Taiwan passport, you will get your citizenship. While living in Taiwan, you can work with your own work permit.
If you are recognized as his daughter, you may be given some legal obligations to take care of him when he will become not able to live on his own.
When you are adopted by him without a proof of your father daughter relationship, your can do naturalization after living in Taiwan for 3 years. While you are living as his daughter, you cannot work, iiuc, so if you want to work, you need to get a work permit sponsored by a company, and should live on a residency based on the work permit. Years on study based residency or blue collar employment based residency are not counted for the 3 years of the naturalization requirment. You need to renounce your Philippine nationality, when you do naturalization, though you can claim it back soon, iiuc.
After you get a passport, you need to get a TARC, on which you will find some information on NIA site.