What qualities or abilities enable Rikki-tikki to fight the cobras to the finish in "Rikki-tikki-tavi" by Rudyard Kipling?

Rikki-tikki-tavi is successful against the cobras because he is curious, brave, and persistent. 

Although all mongooses are supposed to protect people against snakes according to Rikki-tikki, he is particularly adept at the task.  Rikki-tikki has a mongoose’s curiosity, but he is also smart and very brave.  He does not give up until he accomplishes his goal. 

Rikki-tikki washes into the family’s garden and immediately settles into being a house mongoose.  This is every well-bred mongoose’s mission in life.  As house mongoose, it is Rikki-tikki’s job to protect the family and garden from snakes. 

It just so happens that a pair of cobras live in Rikki-tikki’s garden.  In order to have peace, he has to kill both of them. He begins by killing the little dusty snakeling, Karait.  Then it is time to deal with the cobras.  They try to trick Rikki-tikki or scare him, but he is too smart for either. 

Nag is the first to go.  Rikki-tikki fights him valiantly as he is threatening to bite the little boy, Teddy.  After Rikki-tikki kills him, the man shoots him, but Rikki-tikki sees that as pointless.  He was doing fine on his own, even though it was a bit of a struggle against the strong snake. 

The female cobra is now even more dangerous because she is in mourning. She also has a nest of hatchlings.  Rikki-tikki uses his ingenuity to trick her.  He kills all of her babies in their eggs and uses the last egg to lure her out.  

Rikki-tikki has to follow Nagaina into her hole in order to defeat her. 

[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 held on savagely, and struck out his feet to act as brakes on the dark slope of the hot, moist earth. 

The other animals think Rikki-tikki is dead, but he emerges victorious.  With Nagaina gone, Rikki-tikki has successfully dispatched the last snake.  This does not mean he will rest on his laurels though.  Rikki-tikki is vigilant, and will make sure that there are no more snakes in the garden again.