I wrote something about it here.

Short answer: "wiedzieć" is mostly "to know about X", "to know that X", while znać is rather "to know X".

It's not plural here. Like most neuter nouns, "rozwiązanie" has the following three forms identical: Nominative plural, Accusative plural, Genitive singular.

As "znać" takes Accusative (Znam rozwiązanie), when you negate it, it takes Genitive instead (Nie znam rozwiązania). And as you know it's Genitive, it can only be singular here.