Firstly, decide whether you are going to use the Historical Present, or Simple Past tense - your tenses change mid-sentence - see green highlight below:
“The most perfect reasoning and observing machine that the world has seen.” This is how Dr. Watson described Sherlock Holmes. Holmes was dispassionate and cold in his everyday life; however, when he is investigating a case, he shows remarkable passion and intelligence. He was always able to observe the smallest details, analyze them, make his own assumptions and deduce accurate conclusions. However, ‘A Scandal in Bohemia’ is unusual in two ways. Firstly, it shows someone outwitting Holmes. Secondly, the adversary is a woman, and Holmes has previously been somewhat critical of females.
“I am not a whole-souled admirer of women kind,” : be sure to quote accurately!
“I am not a whole-souled admirer of womankind",
he describes himself. The fact that Irene Adler was able to
defeat : defeat is too strong a word. What about (since outwitted was used above) 'outsmart' or 'get the better of'
Holmes using his own tricks,
show : careful - verb does not show agreement with subject, which is 'fact'
strong personality and displays “unexpected” intelligence, which is what attracted Holmes to Irene.
The fact that Irene Adler was able to get the better of Holmes using his own tricks, has such a strong personality and displays “unexpected” intelligence, is what attracted Holmes to her.
Sherlock Holmes thought he is able : again, mixed tenses
to read all criminals’ minds and to predict anybody’s reaction.
to read the mind of any criminal and predict
Are you jumping from criminals, to people in general? and 'reaction' to what? How can anyone predict a reaction, unless that person knows what the 'action' was that prompted the 'reaction'.Do you mean, predict their actions/behaviour.
In “A Scandal in Bohemia,” Irene Adler proved him wrong: she was able to discover his identity and she reacted quickly by leaving the country. The fire (is what-omit) provoked Irene’s suspicion and she knew that Holmes must have been hired by the king to retrieve the photograph. She also had the confidence to follow him to his home in men’s clothes and talk to him without his being able to recognize her.
Irene was the only women in the story who had a personality.
[COLOR="Green"]We all have personalities, no matter how bland. I think you mean, forceful/spirited etc[ personality./COLOR]
Her character unfolds throughout the story and she becomes a significant character.
This needs rewording. A reader's reaction to the first clause is, well of course! - that's what happens in stories. You need to be more explicit as to what you mean. For the second, it is too 'nothing' a statement. You might rephrase it something like, "...and we are taken by surprise that for the first time, a woman becomes so significant a character in..."