I'm sorry, but the chances of JK Rowling letting an amateur, unpublished writer use her ideas is zero. She and her partners have billions of dollars riding on the HP franchise. If she is tired of writing it, she might be interested in some collaboration with a best-selling novel author or screenplay writer she personally enjoys and admires and has worked with before, but not one of us.
Period. She won't even read your letter or know about it, her agents and lawyers will respond with a form letter rejecting you out of hand. That's how the world works with billion dollar franchises.
Get over it. It doesn't matter one iota how much you love it or how well educated you are about the HP universe. Apply your great sequel idea to something else entirely and don't use a drop of anything JKR invented.
I'm not being mean, I'm just trying to save you time, and money, and heartache from being sued or rejected hundreds of times. JK Rowling is not going to collaborate or use ideas from a rando with zero track record. Period.
Read this article, about somebody she did collaborate with for 10 years, Steve Kloves, the screenwriter for nearly all the HP movies: A Screenwriter's Hogwarts Decade. (Summary below) JKR is a micromanager when it comes to her franchise; I am 99.999999% certain she will not let anyone ever write a sequel in her universe. And she's a billionaire with billionaire film industry partners that will sue your pants off if you even try to publish such a thing using any trademark of theirs.
Summary of Steve Kloves' article: Steve began on the first movie, and relates that Rowling told him she understands the movie cannot be the book, but she wanted to stick to her vision of the HP world and its characters as closely as possible. And that the script was something of a collaboration, she corrected him several times on the script, and she had a very deep knowledge of her world that wasn't in the book. When he asked her for the 12 uses of Dragon Breath she reeled them off from memory, even though they were not in the book. And many other such things, in a meeting brainstorming about a scene, he wanted Dumbledore to reminisce about a girl he once knew, Rowling waved him off, because she knew from the first book that Dumbledore was gay. When he wanted a minor character to die and suggested Dobby, she told him no, Dobby plays a critical role in the last book.
Stuff like that. He did write the script, she trusted him to get that and the abbreviated story right; but Rowling is a micro-manager when it comes to details about the HP world and every single character in it, and she won't release her grip on the steering wheel. That makes it difficult for any other writer to tell original stories in the HP world; because the character arcs are already set in stone, in Rowling's mind.