As much as I love redemption stories, they (mostly) aren't believable. Okay, if Han and Leia were both alive and well, they probably could bring their son back without revealing his identity, so people wouldn't want him executed. (You do remember he's killed tons of people? In reality no one would've let him get away with it.) But with Han dead, it'd be unlikely to happen. Even though Leia loves Kylo still, she can never forget what he's done - and she knows he must answer for his crimes. So one day the Resistance captures Kylo. For what he has done in the past, Kylo is sentenced to death. No matter how much Leia loves her son, she can't do anything about that - people demand that he pay for what he's done. The First Order attempts a rescue mission, but for some reason it fails. But Hux isn't ready to give up on Kylo, he just cares too much for him. And so he turns himself in and makes a deal with Leia. He becomes the Resistance's prisoner, tells them everything he knows about First Order's plans and such - in exchange for Kylo's life. In other circumstances, Leia would think twice before agreeing to it, but this is her son, and after she has lost Han, she can't lose Kylo, too. Surprisingly, other Resistance leaders agree to this exchange, mostly because of the valuable data Hux possesses. But after interrogation Hux will be executed - just as Kylo would've been.
Leia lets them say goodbye to each other before the execution. In their last moments together Kylo is completely broken, not wanting to lose yet another person he cares for, and Hux is very calm, comforting Kylo and assuring him it's not his fault and that he is ready to pay this price for him.
Optional, if you feel like it: after Hux is executed, Kylo attempts a suicide, but stops because he remembers that Hux wanted him to live, and if he gives up now it would mean his sacrifice was meaningless. Leia (or someone else?) is comforting him after that.

Yes, I love my ships sad. Make me cry, anon.