I doubt he 'confessed', I suspect they have the evidence he did it, video, DNA and his body is probably marred with the struggle... Then of course there is that laptop which no doubt the authorities have. He probably was offered better conditions and a shorter prison time IF he admitted what the evidence demonstrates...
He most likely figured out he would never leave Peru alive, and his days of terrorizing women are at an end.
He was already fleeing. Caught him in a taxi in Chili and brought him back to Peru.
Yes, I bet he made up a story on the spot. There’s a lot more to it, IMO. He makes it sound like he got a bit miffed and smacked her, accidentally killing her. But that wouldn’t explain the sexual assault, roofies(?) in the car, and and “overkill.”