Dr. Jainway is one of the two Bartok Industries scientists tasked with raising the young Martin Brundle. A thoroughly cold and acidic woman, she has no sympathy whatsoever for Martin's plight and views him not as a person but as a "thing" for Bartok Industries to experiment on.
Consequently, she is one of the adult Martin's least favorite people on the planet. After the slowly transforming Martin escapes, he is found inside of a cocoon at a motel and recaptured. Inside of it, he is slowly turning from human to fly. The cocoon is brought back to Bartok Industries where Bartok tells Jainway to keep an eye on its progress.
Consequently, she ends up being the first victim of the mutated Martin when he inevitably hatches. Jainway is alone in the lab when the cocoon begins to ripple and bulge. Occasionally, she will hear the gurgling noises coming from it, but for the most part she for some reason doesn't think this worthy of investigation. When the fully transformed Martin finally does emerge from the cocoon and attack her, she has nobody but herself to blame for being so grossly inattentive.
How she dies is not seen, as it cuts away to Jainway's colleague Dr. Shepard returning from some other task. When he enters the lab Martin is nowhere in sight, having already left before Shepard got there. There's no sign of Jainway, either. Cautiously searching for her, Shepard finds a huge puddle of blood on the floor, having dripped down from above. Coming to the obvious conclusion, he looks up, and sure enough, a very dead Jainway is (somehow) stuck up to the ceiling.