In what ways do Jerry and the narrator find closeness and strength in their relationship in "A Mother in Mannville"?

The narrator is initially impressed by Jerry's work ethic. Even though he is a small orphan boy, he is able to chop firewood like a man. He proves his trustworthiness in the hard work he executes on her behalf. She learns she can rely on him to chop wood without supervision—she even sleeps in one morning as he works, falling asleep to the "even chopping" of his hard work. The narrator knows can trust him enough to leave her dog with him, even beyond the planned amount of days. When the fog makes returning to the cabin impossible, Jerry rises to the occasion caring for the dog while she is delayed.
This draws the two closer in relationship, at least in Jerry's eyes. The narrator says that she believes time spent with her dog has made Jerry feel close to her. She writes,

He felt that he belonged to me as well as to the animal.

Jerry also displays respect and affection by coming inside to sit with her only when she is not writing; he will not interrupt her work, so he sits outside just waiting for a chance to spend some time with her. On one occasion, when she works so late into the evening, she realizes he has been patiently waiting for the opportunity to spend some time with her; he had waited so long for that chance, the stoop was still warm when she sat on it.
The trustworthiness, consideration, and affection Jerry displays plays a large role in the narrator becoming close to him. Early on in the story, she draws a parallel between Jerry and her father, who she believes also embodied the same rare level of integrity she sees in the boy:

The word that comes to me is integrity. The word means something very special to me, and the quality for which I use it is a rare one.

Jerry displays so much integrity that when he breaks the axe handle chopping wood, he tries to pay for the mending of it himself. He takes responsibility right away, claiming that he let the axe fall wrong, which is why it broke. Ultimately, she pays for the fixing of the axe, but she is impressed by his level of personal responsibility.
The relationship strengthens in trust and attachment as the story progresses, which is why the ending is so shocking. She has come to trust those "grave, gray blue eyes" and has not allowed herself to suspect the boy may become abnormally attached to her for the little bit of deserved kindness she has shown him. Near the end, feeling guilty about abandoning him, she convinces herself that Jerry does not suffer from being lonely, even though he has come every day and worked as hard as possible to try to earn her motherly love. The narrator realizes too late that she let the boy way closer to her than she should have done, given that he was a susceptible orphan boy and she never intended to adopt him.