I'm wracking my brains as to what you're thinking of with 'firing a Prime Minister'.
The Queen has no ability to do so. Maybe your confused that the law requires her to ask for the resignation as formal protocol when a Prime Minister loses confidence of their own political party. As happened with Thatcher.
Personally, by nature, I'm republican in my feeling towards royalty. I would happily see the institution die a death once the current Queen passes on.
However, she herself has done an incredible job in promoting our interest overseas, being a wonderful representative for us and working dilligently for her people for nearly 60 years in the role. She is the spiritual leader of the Church of England and head of our Armed Forces. It is her and her alone that can rule on a declaration of war. She appoints Bishops and Archbishops and raises millions each year for all commonwealth nations with her patronage of charities and institutions and also with her attendance at schools, hospitals and functions. Every day a 'red box' is delivered to her full of documents and reports from government officials and commonwealth ones too. They must all be read and signed by her.
She's 83 I believe now and is still maintaining a near full working schedule. Sure that schedule means she travels, eats and sleeps at a level you and I will never know.....however, I do not begrudge her one piece of it....
Her ass clown husband and children however.......