Are we obligated to continue sponsoring the orphaned family whom my father used to secretly sponsor before his death?

May Allah reward the questioner's father with the best reward for his guardianship of that family. The children are not obligated to take care of this family after their father's death unless he bequeathed something to them from his inheritance. However, if they follow his example and take care of them, they will have a great reward, due to what the Prophet (peace and blessings of Allah be upon him) encouraged regarding the guardianship of orphans.