Every year I burn the chametz for my father (I live at home) because he leaves for work early in the morning, however he still searches the night before
Q: Who says the bracha in the night? My father because he is the owner of the house or myself because I am shaliach to burn the next day and the bracha is connected from night to next day
Your father is the one to say the bracha because he is also doing the nullifying of the chometz at night after he finishes searching for the chometz. The nullifying of the chometz that is done during the day is for whatever chometz he wanted to save, and wasn’t included in his first nullification. The main mitzvah is nullifying the chometz, and burning the chometz, although it is a minhag to do so, in order to fulfill the mitzvah according to all opinions, it is considered an extra mitzvah, and not the main mitzvah. Therefore your father will make the bracha.
As side point, if you are going to nullify the chometz for your father, you don’t say “kol chamira d’ika brshusei…” “all chometz that is in my property, rather you have to say, that is in the property of my father.
O:CH 434-3,4, also 454-1,M:B 434 16,