The lesson from that show wasn’t that the diagnosis was missed - it’s to keep going back if you’re worried, because some things can’t be diagnosed early in the course of the disease.
Kawasaki disease is a syndrome that’s hard to diagnose, and by definition it can’t be diagnosed in less than 5 days and antibiotics don’t help. We use immunoglobulins and high dose aspirin.
http://www.emedicine.com/EMERG/topic811.htm
http://www.aafp.org/afp/990600ap/3093.html
The lesson from that show wasn’t that the diagnosis was missed - it’s to keep going back if you’re worried, because some things can’t be diagnosed early in the course of the disease.