At the risk of being pedantic, let me define sentiment.
1) an attitude toward something; regard; opinion. 2) a mental feeling; emotion: a sentiment of pity. 3) refined or tender emotion; manifestation of the higher or more refined feelings. 4) exhibition or manifestation of feeling or sensibility, or appeal to the tender emotions, in literature, art, or music. 5) a thought influenced by or proceeding from feeling or emotion. 6) the thought or feeling intended to be conveyed by words, acts, or gestures as distinguished from the words, acts, or gestures themselves.
The piece as a whole is sentimental because of the relationship between this lonely demented man and his kind neighbors, also because of the theme of self sacrifice.
The Sentimentality of the piece is woven throughout. Edvard's every interaction with the little girl and her mother is sentimental. His thought's about them as he plans his excavation are sentimental. The theme of self-sacrifice that threads through the piece is sentimental. I'm not sure if anything except the statistics, is unsentimental. Even his attachment to comic books is sentimental in a geekish sort of way.
That clashed with the genuine if somewhat cliche'd emotions of the piece.
Why does the affected tone of a hard-boiled narrator clash with the sentimentality of the piece? Because the sentiment is ingenuous (not ingenious) and open-hearted, even naive. While the narrator tries to be calculated and even a touch cynical. It clashes because the narrator doesn't sound like he is reading the same story we are, and what he is picking up he doesn't quite understand.
How is the emotion of the piece cliched? It is boiler-plate chick-lit fiction. The stereotypes are so thick you have to walk carefully to avoid cutting yourself on them. We have the
Martyred Messiah,
Self Discovery Through Battling the Figments of One's Imagination (Quixote),
The Eternal Apprentice (in the little girl), and the
May December Romance in a very chaste version. Also the more modern archetype of the "
Concerned but Ultimately Ineffectual Neighbor".