It means "after giving it enough thought." To "reflect" can mean to think seriously, often about something that already happened. And "due," when used in front of a noun, means that something or someone was "due," or they were owed, or they deserved, a certain amount of the noun. For example: "I treated her with due kindness" means I treated her kindly because she deserved it. Or, in the legal phrase "due process," this means that all people charged with a crime are owed fair legal treatment.