A 68-year-old woman with a history of left-sided breast cancer (treated 5 years ago) now presents with progressive hoarseness and a weak, breathy voice. On laryngoscopy, the left vocal cord is seen in a paramedian position with poor abduction. Additionally, the examiner notes that the left vocal cord appears shorter and sits at a slightly higher level compared to the right. Which nerve is injured, and what is the anatomical explanation for the cord appearing shorter?