Is the sentence correct

Sure. I would say there is one hyphen too many.

Is the doer missing? Should it be "By the time someone had put out the flames with a fire extinguisher, she had third-degree burns that covered 60 per cent of her body".

I think 'she' is quite acceptable. Since the sentence does not exist in a vacuum, I'm sure we know who 'she' is. Even if we don't, it's still a correct sentence.