I think the first thing you both need to do, though, is to adjust your attitudes in the situation. I know this sounds a bit accusatory, but hear me out. You describe Emma, who is a child, as a "spoiled brat." You don't just see her behavior as bratty, but her personality as bratty and somewhat irredeemable. Children can pick up on when they aren't liked, and-like it or not-when you married Diana, you also made a commitment to Emma that includes being there.
My close friends have several kids who are generally "OK" as kids. I've had some fun conversations with the kids as they've grown but my issue is one of the kids ("Alex") was babied (parents acknowledged this) and her behavior has really gotten out of hand. Think weaponized incompetence, neediness, unhelpful most of the time accompanied by extreme attitude and yelling at the parents when asked to do something she doesn't want to do, often relies on their other siblings to do things for them-and she's the oldest!
In a study published in the journal Royal Society Open Science, younger children were found to be slightly better at delaying gratification than older children, highlighting age-related differences in self-control.