In A Short History of Nearly Everything, who has the power? How was the power achieved and maintained? I'm a parent trying to help my child with this book.

The starting point here is to explain that Bill Bryson is a popular journalist and travel writer trying to write a travelogue of ideas and science. He writes with a journalist's eye for anecdote, and many readers find his accounts of scientists, as opposed to their discoveries, the most riveting elements of the book. 
A useful way to think about how power functions in the book is to analyze the "great man" version of the history of science. Consider the major discoveries in the history of science. For each discovery, you would investigate how we tie that discovery to an origin story centered on a "great man": Einstein and relativity, Newton and gravitation, Copernicus and the heliocentric solar system, Darwin and evolution, etc. Even though Bryson often debunks "great man" and "great moment" foundation stories, emphasizing the accidental nature of discovery and the human quirks and foibles of discoverers, his history of modern science is one of people, and people who are overwhelmingly white, male Europeans. 
As you consider, for example, the discovery of cosmic background radiation, you might ask how many other people who are not named or even suggested by the book are involved in the discovery. Think about who manufacturers the equipment, maintains the facilities, and funds researcher salaries or the armies of graduate assistants and lab technicians involved in academic science. As you think about this, you can address power in two ways. The first would be to work through the book thinking about the names that are not mentioned and the degree to which science requires vast amounts of collaboration. By recounting science as anecdote Bryson recapitulates an inequality of power and credit within the scientific world. Next, as you look at various scientific discoveries, you might think about how scientific knowledge and discovery are enabled and funded by wealth and power and serve to replicate it. Scientific discoveries and technological advances create a virtuous cycle that benefits certain nations and classes and may contribute to a widening of inequality. The country with an atom bomb is far more powerful than the country with only daggers and spears.