The process took several months, but mostly because I was still working on my case study and interviewing elsewhere. I studied, practiced and prepared a LOT for this interview and it was still difficult. The difficulty is partially natural for being in UX; it takes many hours to craft a compelling case study and update your portfolio. The other part of the difficulty was in nature of being specific with your examples, leading them back to the leadership principles and also highlighting how you match what they are looking for. However, every single person that interviewed me was kind patient, they know how high of a standard Amazon interviews are.
My advice; practice your case study in other interviews before you do the full loop. Research the heck out of the leadership principles and decide on about 8 stories that cover all of the principles, it's okay if one story highlights multiple principles. Also - when they dive into your story that is normal. They are supposed to ask follow up questions to pick at your answer, it's a good sign not a bad one like you didn't give a good story. Lastly, do the extra hour long prep call with your recruiter where they give you tips and tricks for the loop. They are there to support you to PASS the loop. They have a lot of insights and you can do both dry runs of your case study presentation (which I did) and you can share your questions for the interviewer (which my recruiter critiqued and gave me a few extra).
Last tip: if they ask you what you would change about XYZ, always have an answer and be honest/positive. Never say, "nothing" because it shows you don't learn/grown.