How did Rikki-Tikki kill Nagaina in "Rikki-Tikki-Tavi" by Rudyard Kipling?

Rikki-tikki killed Nagaina by following her into her hole and fighting her to the death.

Nagaina was the female cobra in the garden at the Indian bungalow.  Rikki-tikki killed her mate, Nag, earlier in the story.  However, it turns out that Nag and Nagaina had eggs.  Rikki-tikki knew that he had to kill all of the snakes in the garden for the family to be safe, so he killed all of the eggs except one and used that egg to lure Nagaina out of the house.

Nagaina is described as “Nag's wicked wife.”  She certainly seems to be the planner of the two.

"When the house is emptied of people,'' said Nagaina to her husband, ``he will have to go away, and then the garden will be our own again….”

Nag wonders if there is anything to be gained by killing the people, and Nagaina convinces him that they won’t survive as long as the humans are there.

Rikki-tikki describes Nagaina as “worse than five Nags” when she finds out that he has killed her babies.  She does indeed threaten the family, and Rikki uses the last egg to lure her outside.  The other animals assume that once he follows her into her hole he is dead, because a cobra hole is very dangerous.  Yet Rikki-tikki emerges alive.

Rikki-tikki kills Nagaina by cunning and skill.  He is smart enough to save one of the eggs as an incentive for her to follow him.   He also is able to kill her even in his hole.  This demonstrates not only his bravery and devotion, but his luck and ability.

How does Rikki trick Nagaina or Nag in "Rikki-Tikki-Tavi"?

Rikki-Tikki-Tavi tricks both snakes, Nag and Nagaina, so that he has the advantage when killing them.

Rikki tricks Nag by waiting in the house while Nag hides in the bathroom, so that he will be able to bite him in the right place and kill him without being in as much danger.

"Now, if I kill him here, Nagaina will know; and if I fight him on the open floor, the odds are in his favour. What am I to do?'' said Rikki-tikki-tavi.

Rikki-Tikki-Tavi tricks Nagaina by killing all of her eggs and running off with the last one so she will have to follow him.

Rikki-tikki put his paws one on each side of the egg, and his eyes were blood-red. ``What price for a snake's egg? For a young cobra? For a young king-cobra? For the last -- the very last of the brood? The ants are eating all the others down by the melon-bed.''

Rikki-Tikki gets Nagaina mad, so she is not thinking clearly, and she does what he wants her to do so that he has the advantage.  It does not work quite as he expects, and he has to follow her down the hole, but he emerges triumphant.

How is Nag killed in "Rikki-Tikki-Tavi"?

After Nag and Nagaina meet Rikki-tikki-tavi in the garden, they formulate a plan to get Rikki-tikki to permanently leave the bungalow. That night, Chuchundra the muskrat informs Rikki-tikki that Nag and Nagaina are plotting something wicked, and Rikki-tikki overhears the two cobras discussing how they will kill Teddy's father.

Nag proceeds to crawl into the bathroom sluice and waits for Teddy's father to use the bathroom, where he plans on biting him. While Nag is sleeping, Rikki-tikki sneaks up behind him and pounces on his head, biting down with all his might. Nag attempts to whip Rikki-tikki off his head but Rikki-tikki's jaw remains locked. Despite being viciously trashed throughout the bathtub and almost losing consciousness, Rikki-tikki refuses to let go. Teddy's father is awakened by the noise, grabs his shotgun, and unloads both barrels into Nag's massive hood.

Overall, Nag is killed when Rikki-tikki ambushes him in the bathroom and Teddy's father proceeds to unload his shotgun into Nag’s hood.

How does Rikki trick Nag in "Rikki-Tikki-Tavi" by Rudyard Kipling?

With Nagaina's new nest of eggs nearly ready to hatch, the two cobras knew that the bungalow must be cleared of humans in order for their own babies to survive. Nag decided to wait in the bathroom for the man to appear the next morning, curling himself up in the water jar that he knew would be used once the man had awakened. But Rikki was watching, and he decided to attack Nag in the large jar while the snake was still asleep: It would be his best chance to kill the large cobra without being killed himself.

"If I don't break his back at the first jump," said Rikki, "he can still fight; and if he fights -- O Rikki!
     "It must be the head," he said at last; "the head above the hood; and when I am once there, I must not let go."

Rikki attacked, and he did not let go. He locked his jaws, knowing he would be battered to death by Nag's relentless whipping, but he was willing to die "for the honour of his family." But the man had awakened, and he put an end to Nag with a blast from his shotgun. The man saw that Rikki was a hero yet again:

"It's the mongoose again, Alice; the little chap has saved our lives now."