I think you might be worried about nothing.
Let's put the shoe on the other foot. You're trying to find a baker to make your wedding cake. You've sent a few emails back and forth with 'ABC Pastries' but you eventually decide to go with someone else. You send them an email that says, "Sorry, but I've decided we're going to have XYZ Bakery handle the cake."
... do you honestly care whether ABC Pastries sends you a reply? From your perspective, the matter is done. You've made a decision and notified the appropriate party. Sure, ABC might send a 'thanks for your consideration' email, but... it's not something that has a whole lot of meaning to you at that point. All you'd realistically care about is whether ABC is going to send you some piece of information that would make you change their mind.
So Candidate K declined your offer - from their perspective, the matter's settled. Unless you're going to provide something to change their mind, there's not really anything they need from you. Sure, go ahead and send a 'Thanks for your consideration' or such, but... it's not going to matter all that much from their side.