Will one be obligated to say Kaddish for a sibling who has rejected all connection to Judaism, married a non-Jew, is raising her children with no religion whatsoever, and has insulted her brother’s evolving and growing Mitzvah observance?

Your question is not a simple one. Most Poskim today consider non religious Jews, even those who have intermarried, as "Tinok Shenishba", literally a captured child, that is, one who was not exposed to Jewish values and therefore cannot be held responsible for rejecting them. According to this position and reasoning, all fraternal obligations would apply, including Kaddish. If the brother had Jewish upbringing and then rejected Judaism the position might be different, but this requires consulting a competent Posek on the particulars of each case. It is difficult to fulfill mourning obligations for one who has insulted and belittled you. It sometimes happens that in the middle of a divorce process, one of the spouses dies. The Halacha clearly calls for observing the rituals of mourning, even though emotionally this is stange and the emotional commitment is lacking. Again, I urge you to consult a Rabbi about the particulars of your upbringing and their ramifications.