Instead of creating new element, you need to select the existing element, for example :
var existingRoot = xmlDoc.SelectSingleNode("//music");
Then you can do exactly the same way as you did to add new <song>
element :
var childTag = xmlDoc.CreateElement("song");
existingRoot.AppendChild(childTag);
var childertag = xmlDoc.CreateElement("name");
childTag.AppendChild(childertag);
var childertag2 = xmlDoc.CreateElement("singer");
childTag.AppendChild(childertag2);
var childertag3 = xmlDoc.CreateElement("chords");
childTag.AppendChild(childertag3);