It's certainly a common literary trope by the mid-20th century that pleading insanity at trial is an easy way to avoid punishment for a serious crime: the idea seems to be you will be sentenced to a mental institution and then released swiftly once "cured". That's certainly not how legal insanity works these days, but is there a historical precedent for someone successfully and outrageously gaming the system in a infamous case that led to its ubiquitous literary use, or is its prevalence simply a literary device feeding on itself?
Probably the most high-profile successful insanity defense was probably too late to inspire the literary trope, but was nevertheless very important: the case of John Hinckley Jr., who shot President Ronald Regan in 1981. Hinckley was obsessed with actress Jodi Foster and apparently believed that as an assassin of a president, he would be a public figure and therefore her equal. The jury in his criminal trial found him not guilty by reason of insanity.
There was a fairly substantial backlash to the verdict--how could a man who shot the president be not guilty? A number of legal reforms resulted. As an example, in 1984, the United States Congress passed the Insanity Defense Reform Act, which applies to federal crimes (for those not familiar with the US legal system--most criminal offenses are state-level, so this doesn't directly control, for example, your average murder trial. A number of states also changed their laws, though). This act required the accused to prove insanity by clear and convincing evidence rather than requiring the government to prove the accused sane, as under prior law. The act also prohibited experts from testifying that the defendant was sane or insane--they can only talk about the mental diseases. Finally, it limited the defense to people who can't appreciate the nature and character of their acts. Under prior law, and the law of some states, people could also make the insanity defense if they essentially knew what they were doing and couldn't stop themselves (so-called "irresistible act").
Speaking of iressistible acts, one literary example of the defense being used as an escape is the book and film Anatomy of a Murder. I've never read the book, but the film is fantastic and deals directly with the insanity defense generally and the "irresistible act" prong specifically. The book was written by then-Michigan Supreme Court judge Paul Voelker, so the story has a lot more legal grounding than the usual courtroom drama. The reason I mention it is that it was based on a real trial defended by Mr. Voelker as an attorney. So while that real trial wasn't necessarily so high profile that it caused the trope because of a general anti-insanity defense fervor it was a real case that inspired one instance of the trope.