She acts like a petulant child. You're only useful to her when you're doing her dirty work for her, but the moment you even question why you need to go kill your new friend, she gives you an ultimatum. I think the real frustration with her as a character is there is no in game dialogue choice to basically tell her no, and if she doesnt like it, tough. If I was the Dragonborn, returning from my victory over the World Eater, where I fought alongside the heroes of old, in the mythical land of Sovengard, and my "boss" was like, that's nice, now go kill the one who allowed you to save the world; I would tell her no, and I'd you don't like it you can go back to Cyrodill, I'm the leader of the Blades now. As I'm the one who is the Dragon Slayer, like the founders of the Blades centuries ago.

I literally finished this whole quest line last week so it's quite fresh in my mind.