Miss Dent's handwriting is significant because it serves as a clue that was greatly missed by Blake. Blake had given Miss Dent an opportunity to work for him as a secretary when she applied to work for him
"after so many months at the hospital".
Blake, an antisocial and somewhat narcissistic guy has very little knowledge of the world about him because he lives from the facade that he has created of himself as a sophisticated, larger than life man who thinks that he is better than everybody else.
After he has a one night stand with Miss Dent, he feels bored as usual and decides to fire her, using as an excuse that her handwriting is "undisciplined". After firing her Blake starts to wonder about Miss Dent, because she begins to follow him around, showing up just about everywhere he goes sporting a "contorted" face that he scoffed off judging her as a "shakeable" girl, or someone he could override in case she goes a bit crazy.
Yet, as he puts the dots together, he realizes that people who have such handwriting could very well be "erratic" or not very well in the head. Moreover, he wondered what Miss Dent meant when she thanked him for hiring her after "so many months at the hospital". He realized that he never asked her what sort of hospital it was or what injury she had sustained..then he realizes that Miss Dent might be a mentally ill woman.
This is indeed the case, and the handwriting was a very open clue that, had he been more observant of others and less into himself, he could have picked up on it and save himself from his current situation.