Sidious tells Rey to strike him down in anger so his soul can inhabit her body, and the two can become one. Yet, when Rey does kill him, he doesn't do this. That's simply because she had to do it in anger. This is the way of the Sith, and by killing him in revenge for the deaths of her parents and her Resistance friends, she would become one of them. Palpatine could then transfer his soul into her. However, she refuses. Palpatine then drains the life force from her and Ben Solo. Since they're a dyad in the Force, it fully heals him, allowing him to stand and wield the Force without needing a new body. When Rey stands again, with the help of the Jedi from generations gone, Sidious no longer needs her, so he tries to kill her with his all-devouring Force lightning. Using the lightsabers of Luke and Leia, Rey deflects his lightning back at him. She doesn't instigate the attack (the sign of the Dark Side - as she has done throughout the film so far) and simply reflects the Dark Side back at her grandfather. Thus, his soul has no Sith to transfer to when he dies. He's definitely dead this time. Probably.