Rikki fights Nagaina and emerges victorious after killing her.
Cobras do not likely have many natural predators. The mongoose appears to be one. The conflict between Rikki-tikki and Nag and Nagaina is evident from the first time they meet. Mongooses kill snakes, and the cobras are obviously afraid of having Rikki around.
Nag was thinking to himself, and watching the least little movement in the grass behind Rikki-tikki. He knew that mongooses in the garden meant death sooner or later for him and his family, but he wanted to get Rikki-tikki off his guard.
When Rikki-tikki first meets Nag, the male cobra tries to distract him so that Nagaina can kill him. Rikki-tikki is at first not aware of that fact that there are two cobras. The cobra is a formidable foe, even for a swift mongoose. The two of them try to tag team him. It doesn’t work. Rikki-tikki is smart enough to avoid the snakes.
In fighting the cobras, Rikki-tikki kills Nag first, and then Nagaina. Nag comes into the house, where he is also a threat to the people. Rikki tries to be more careful with Nagaina. He develops a plan, wherein he kills all of her baby cobras except for one, and uses it to lure her away from the house. Then, he is able to keep the people out of harm’s way.
Rikki-tikki is aware that following a cobra into her hole is a bad idea.
[Very] few mongooses, however wise and old they may be, care to follow a cobra into its hole. It was dark in the hole; and Rikki-tikki never knew when it might open out and give Nagaina room to turn and strike at him.
He does it anyway because he knows that he has to get her while he has the chance. Everything has led up to this. He killed Nag. He killed the baby cobras. He got her confused and frustrated and angry and led her away from the house. Now she has the upper hand, but he makes sure not to let her take advantage of him.
Rikki does emerge victorious. He is able to kill Nagaina in the hole. The other animals are sure that she will kill him, because the situation is so dangerous and they cannot see what is happening down in there. Darzee even goes so far as to compose a song. Yet Rikki-tikki-tavi lives, and he is basically unhurt—just tired.
The fact that Rikki-tikki would follow Nagaina into her hole shows that he is both brave and brazen. There is a touch of foolishness in his actions, but also calculated risk. He is confident enough in his skills, and in the fact that he has the upper hand. She is not thinking clearly, in her anger and her grief. He acts with cool calculation and very little emotion. He is confident he will win, and he does.