How to apply for guardianship for my father with Huntington's Disease?

You can be guided by your father's attorney in applying for guardianship. You say he is currently staying with you. Does he intend to continue to live where you are? That would require new doctors, new residency, and etc. His funds can pay for the guardianship. If the siblings are problematic I would then not include them in your actions, lest they fight for guardianship over your father on their own, and in their own state, where I assume your father lived.

You contact a lawyer. Could be an Elder Care Attorney or you could see one that does Family Law.
Is your dad agreeing that you need to become his Guardian?
Guardianship has it's problems and if you can avoid it it might be easier for everyone. Guardianship can also be expensive and it does require visits to court and detailed recordkeeping as far as expenses.
It might be possible that with the documents you have you might not need to become Guardian. (not sure if the 5 Wishes is detailed enough.) But it might be worth asking. But you need to at least discuss this with a lawyer.