Varför valde Voldemort en orm som Horcrux med tanke på att det är en levande sak och har en begränsad livslängd?

Vi vet inte riktigt. Vi har bara vad Dumbledore berättade för oss:

"I don't think so," said Dumbledore. "I think I know what the sixth Horcrux is. I wonder what you will say when I confess that I have been curious for a while about the behavior of the snake, Nagini?"
"The snake?" said Harry, startled. "You can use animals as Horcruxes?"
"Well, it is inadvisable to do so," said Dumbledore, "because to confide a part of your soul to something that can think and move for itself is obviously a very risky business. However, if my calculations are correct, Voldemort was still at least one Horcrux short of his goal of six when he entered your parents' house with the intention of killing you.
"He seems to have reserved the process of making Horcruxes for particularly significant deaths. You would certainly have been that. He believed that in killing you, he was destroying the danger the prophecy had outlined. He believed he was making himself invincible. I am sure that he was intending to make his final Horcrux with your death.
"As we know, he failed. After an interval of some years, however, he used Nagini to kill an old Muggle man, and it might then have occurred to him to turn her into his last Horcrux. She underlines the Slytherin connection, which enhances Lord Voldemort's mystique; I think he is perhaps as fond of her as he can be of anything; he certainly likes to keep her close, and he seems to have an unusual amount of control over her, even for a Parselmouth."
Harry Potter and the Half-Blood Prince, chapter 23 (emphasis mine)

Så han gjorde henne Horcrux eftersom han fortfarande behövde en, och han gillar henne.