There isn't a drop of this stuff left in the world - whatever wasn't destroyed was almost definitely consumed. That makes it difficult to break down exactly what went into Starrick's Soothing Syrup, but we do have some clues courtesy of Mr. Charles Darwin.
The first ingredient Darwin mentions, datura stramonium, is known to cause hallucinations, increased heart rate, and an aversion to sunlight. It was once believed that Devil's Snare was capable of turning people into mindless and utterly compliant zombies. Hence the plant's appeal to Starrick.
The other ingredient that we know of, opium, is... well, it's opium. The drug is so addictive and debilitating people created dens for the sole purpose of consuming it and spacing the heck out. Even if the only other ingredients were vitamin C, sunshine, and rainbows, Soothing Syrup would still leave you worse off than before you drank it.
It seems as though Starrick's Soothing Syrup lived and died in 1868. No advertisements or articles about the tonic exist before that year, and hardly any exist after that year.