Вам не нужно создавать новый StreamReader, просто используйте GetResponseStream
//Get the Response Stream from the URL
Stream responseStream = response.GetResponseStream();
// Read the Response Stream using XmlTextReader
XmlTextReader reader = new XmlTextReader(responseStream);
//read through all the nodes
while(reader.Read())
{
//find the item node and read its value
if(reader.Name == "item")
{
Console.Write(reader.ReadString());
}
}