O. Henry never goes into elaborate, flowery, romatic details on their emotions for one another. There is nothing super cheesy or dramatic. There are statements, simply put, here and there, that relay the depth of their emotions, but they are never overdone. For example, O. Henry writes of Della's affections for Jim, that "Many a happy hour she had spent planning for something nice for him." So, the time that she spent planning on getting him something, was a happy time; she loved him so much that her sacrifice for him was happiness. Henry goes on, stating that Della wants to get "something...worthy of the honor of being owned by Jim." She feels Jim a very wonderful person, and to be owned by him is an honor. So, those statements relay her depth of emotion.
For the most part however, O. Henry lets the actions of the characters portray the depth of their love for one another, rather than words. Della is willing, with barely any hesitation, to sacrifice her most prized possession-her hair-to get something nice for Jim. And Jim, is likewise willing to sacrifice his precious watch for her. It is not that they want to show off how wealthy they are, but that they want to give a token of the depth of their love. So, O. Henry shows the depth of their emotion through the great sacrifice that the two characters make. They show their love in action and selflessness.