If your father lived in New York, there may be limits on how much information you have access to because of privacy restrictions. I think this is a case where you will be much better off hiring a professional to help you, especially since you live outside of the USA. There are several questions on this site about doing research in New York, and some on hiring a professional, which you can look at for ideas.
If you do want to hire a professional, you will need to have all your information in order, so you can present a summary of what you know and where that information comes from. Here are some things you can do to get ready:
- Make a list of all documents that you have collected, like the 1943 marriage certificate that you mentioned in your question. Put this list in chronological order.
- Make a timeline of all the information you have about your father, referring to your source list to show where that information came from.
- Write a biographical sketch of your father, including the information from the previous two steps.
- Make a summary of the searches you have tried so far which have been unsuccessful. Note where you searched, and how you searched (for example, write down what spellings of the surname you tried to look for).
- Make a To Do list of things you have not tried yet (ideas from other questions here).
- Write a short introduction of yourself, with some information that you want your half-sister to have, that the professional can give to your half-sister or her family.
In the bottom of my answer to the question Discovering descendants of 19th Century German immigrant to Australia who served in World War 2? there is a list of resources for people who want to trace the descendants of a particular person -- look for the links under Resources. There may be search techniques you have not tried in the how-to videos and guides.
You might also consider taking a DNA test and consulting with a genealogist experienced in DNA testing to help you in your search. You might be able to find matches that way, if there are descendants of your half-sister who are looking for others related to your father.
I agree with Fortiter's caution -- it is important to keep in mind that your half-sister and her family may not want to make contact with you.
Think about what would happen if someone contacted you out of the blue and claimed to be a relative. It might be easier for some people to be approached by a third party, someone whose reputation is known, and whose credentials could be checked, before making contact with you directly.