This is "The End of the Whole Mess", indeed by Stephen King.
From Wikipedia's description:
The story, narrated by Howard Fornoy in the form of a personal journal, recounts the life of his genius younger brother, Robert. Bobby, a child prodigy whose adult interests led him to study a variety of scientific disciplines, discovered a chemical that reduces the aggressive tendencies of humans and other organisms. While doing sociological research in Texas, Bobby used crime statistics to create a sort of topographic map which displayed a geographical pattern of violent crime. Examining the map, Robert noted diminishing levels of crime centered on the town of La Plata. When he arrives to investigate, he finds that this town has never had any violent crime.
Several months later it is discovered that, to the Fornoys' horror, there was another constant about La Plata that was not studied until after the substance was released. It does eliminate aggression, and increases calm, but it does the job too well. It builds up, out of control, in a subject's system, ultimately giving them symptoms resembling dementia or Alzheimer's disease and eventually resulting in death. Howard's journal entries after this point begin to include increasing amounts of grammar, spelling, and other mistakes, eventually devolving into incoherence as Howard succumbs to the effects of the chemical. It is implied the human race will also eventually die out as adults start to forget how to care for newborn children.
The last part of the story is:
I have a Bobby his nayme is bruther and I theen I am dun riding and I have a bocks to put this into thats Bobby sd full of quiyet air to last a milyun yrz so gudbo Im goin to stob gadbo bobby i love you it wuz nt yor fait i love you forgiv yu
And a childlike scrawl of a signature.