I'm going to have to be on the "this does not fit the characters" side of the argument. Throughout the show, Konata demonstrated herself to be pretty well-balanced for someone who would be considered a social misfit. I heavily doubt that she would care if no "normals" talk to her because she would very likely start hanging out with other otaku (see Patty and Tamura). Her interests are not out there enough for there to be few to no other people with similar interests, and she certainly seemed to get along well with her workmates in the maid cafe. No, Konata has too many potential supports to be a cutter, even taking into account the mind-numbing blow having a potentially fatal condition may be.
Kagami, on the other hand, is possible... if you assume her to be 100% irretrievably yuri for Konata. But she has consistently demonstrated an interest in boys and there are definitely men who like the strong-willed, confident image she always puts up in public, so it's not likely that she will never have anyone asking her out or rejecting all of them.
Characterisation issues aside, however, the story itself is very well-written. It touches the people, even those who have an issue with it. Be honest; some of us bring up those characterisation issues *because* we don't want to see the characters in such straits.