How do I write an analysis and thesis for John Updike's "Dog's Death"?

Updike's poem "Dog's Death" is about the unseen hurts that creatures we love walk around with and that they cannot survive even though we love them. That is a thesis about the poem. After his puppy dies, the narrator finds out that the dog had been walking around with a ruptured liver. All the time that the family was trying to encourage the puppy to follow rules, such as using newspapers to go to the bathroom and praising her with the words "good dog," the dog was carrying around an injury, perhaps from a brush with a car or a kick, that will kill her. After the dog dies, the family realizes she really was a good dog and had been suffering with diarrhea but still dragged herself over to use the newspaper. Walking around with this unseen injury, she tried to be a good dog until the end of her life. The narrator says, "Though surrounded by love that would have upheld her,/ Nevertheless she sank and, stiffening, disappeared." The narrator is crushed by the idea that his dog died, even though she was loved, and he realizes that love was not enough to help the dog survive.
This poem is not only about the unseen injuries that people and animals carry around in spite of being loved but also about the way death sometimes accompanies life. Another thesis for this poem might be that death is always present, even in the midst of youth. The puppy is ironically moving towards death even as the family is encouraging the dog to embrace the life of a young and healthy puppy. Updike writes, "As we teased her with play, blood was filling her skin/ And her heart was learning to lie down forever." While the family encourages the young dog to play, she is already dying. The family did not understand that in the midst of their young dog's life, death already loomed.