"Ein jeder" und "jeder" are synonymous in their meaning, but putting "ein" before "jeder" makes it substantival and puts some emphasis to fact that a condition or circumstance applies to each and every one.
Google translates the phrase to "eines jeden Kindes" and I as a German definitely side with that when it comes to correct grammar. "eines jeden" is simply the genitive of "ein jeder/ein jedes". "jedes Kindes" is grammatically correct, it is the genitive of "jedes Kind", but unusual in every day language.
Colloquially one would say "von jedem Kind" because in current German language the dative is the genitive's death.
"... der alle Kinder ..." is wrong sentence building.