The ending to Revenge of the Sith was kind of weird. His whole reason for going to the Sith was to save Padme, who didn't survive anyway. So why would he keep Palpatine as a mentor.
At the end you see them working together on the Death Star.
Anakin: "Where's Padme."
Palpatine: "You killed her"
Anakin. "Impossible. She didn't survive? Nooooo." Then he's still Sith
I think you're over simplifying the reason for Anakin's change.
Yes he wanted to save his wife from death, and Palpatine offered that. However Anakin's real reason for falling goes back as far as the Phantom menace, where he told Qui-Gon that he dreamed that he would come back and free the Slaves.
Then later when his mother was killed and he could do nothing about it.
Once Anakin came to the Jedi Order, Palpatine kept pushing in those tap roots.
Hey Ani, the Jedi are limiting your power, but I see you being the most powerful Jedi and they need you because of that.
Remember that the Jedi were basically limiting him and Palpatine was lying to him with the truth.
They don't trust you.
They are frightened of your power so they try to limit you
On top of that, he promised nothing more then power. That whole speech in the opera house wasn't about saving Padme, it was about Anakin gaining power that he couldn't get from the Jedi, that he was bound in chains by the Jedi.
So when it came down to his speech, it was mainly. Hey look Ani, the Jedi are conspiring against the Republic, they want to limit your power while increasing their own. You've always dreamed of doing great things and becoming a man of conscience, that can't happen if your with the Jedi, and by the way, they're lying to you and conspiring behind your back to kill the only one that's shown you respect and that's me. Oh and by the way, I'm that path to power and I might have the power to save your wife.
So Anakin who's already angry at the Jedi and feel they're pretty much corrupt and goes back, and Mace basically says, I don't trust you and I'm going to see Palpatine.
So in effect everything that Palpatine said is true from a certain point of view.
Anakin wanted power more then anything else he even said that to Padme who he was seeing as more of a possession at that point then anything else.
So why didn't Vader slaughter Palpatine after his fall, I mean he talked about it with Padme. But in the end, Palpatine was the only one who ever really understood him, accepted him and told him the truth. He also knew that he needed Palpatine for power. Also Anakin was born a slave, and ended up a slave and he was willing to accept that as penance