Here are 2 prototype-functions I created.

[HaganeNoKokoro]

Your way of figuring out the fraction is definitely better. I just pulled that other one out of my @$$, although it's aiming for the same "something/999" kind of thing.

Here's what I mean about marking up the string. I tell the function which part of the decimal repeats by putting it in square brackets

String.prototype.toRational=function() {
  var a=this.substring(0, this.indexOf("."));
  var b=this.substring(this.indexOf(".")+1, this.indexOf("["));
  var c=this.substring(this.indexOf("[")+1, this.indexOf("]"));
  
  var lena=a.length;
  var lenb=b.length;
  var lenc=c.length;

  var x=parseInt(a+b);
  var y=parseInt(a+b+c);
  
  var z=y-x;
  var den="";
  for(var i=0; i<lenc; i++) den+="9";
  for(var i=0; i<lenb; i++) den+="0";

  return z+"/"+den+"="+z/parseInt(den);
}
document.write("5.1[42857]".toRational());