We use "no" to negate a noun (including its determiners and adjectives), in this case, "no (ordinary) girl". This is perfectly correct; it is not slang. We use "not" to negate a verb, so we could rewrite the sentence using "was not": "This was not an ordinary girl." Both sentences have the same meaning; it's just a matter of style.

"She was no ordinary girl" is simply another way to say "She was not an ordinary girl." I respectfully disagree with Stephanie. It is not slang and is grammatically correct. It is simply a slightly more emphatic or dramatic way of saying something. No + adjective + noun Examples: The boxer had no worthy opponents = The boxer didn't have any worthy opponents. The candidate had no strong supporters = The candidate did not have any strong supporters. No + noun Examples: She was no expert = She was not an expert. There was no class on Monday = There wasn't a/any class on Monday. You can also use this sentence structure as a form of insult: He is no Einstein = He is not very smart. She is no spring chicken = She is not young (idiom used mostly by older people)

"No ordinary" is a common usage; just now on Google a search for the exact phrase produced well over 7,000,000 matches. The Cambridge English Dictionary refers to it in its entry linked below. This includes " Her last concert appearance in Britain was no ordinary (= a very special) performance." https://dictionary.cambridge.org/dictionary/english/ordinary And the Ngram Viewer shows that "no ordinary girl" is a little more popular than "not an ordinary girl" - see https://books.google.com/ngrams/graph?content=+not+an+ordinary+girl%2Cno+ordinary+girl&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cnot%20an%20ordinary%20girl%3B%2Cc0%3B.t1%3B%2Cno%20ordinary%20girl%3B%2Cc0

It was a slang usage. It was incorrect and meant to be a little funny in its incorrect-ness. It was laughing at one's self.