Take another look at the sentence:
To finish by the 3 p.m. deadline, an infinitive phrase, has no target in the sentence to describe. We find Sylvia's fingers, but not the typist herself. When a modifier has no word to describe, it is dangling.
To fix the problem, you should revise the sentence like this: To finish by the 3 p.m. deadline, Sylvia made the keyboard sing with her flying fingers.