WASHINGTON—For more than a year, American diplomats have held secret talks in Pyongyang and European cities with North Korea’s top nuclear negotiator, hoping to free U.S. prisoners and even establish a diplomatic channel to constrain North Korea’s nuclear and missile programs.

The official dispatched by North Korean leader Kim Jong Un—Madame Choi Sun Hee—is well known to U.S. officials, fluent in English and is believed to…