Converting a JSON object to XML to be passed to an ASMX webservice
Use camelCase to name local variables instead of words separated by underscores. Although there are no guidelines for them it is common practice among C# programmers.

You should always call XmlWriter.Create to get XmlWriter instead of creating XmlTextWriter.

StringWriter, XmlWriter and any other writer implement IDisposable and thus should be used within using section:

using (var stringWriter = new StringWriter())
using (var xmlTextWriter = XmlWriter.Create(stringWriter))
{
    // ... do your stuff that requires stringWriter and xmlTextWriter
}


You should try to avoid .Result for async operation. Try to mark your method with async and use await for client.PostAsync.

Creation of XML looks awful... I suggest to extract your request XML template into XML document with kind of placeholders like:

&lt;?xml version="1.0" encoding="utf-8"?&gt;
&lt;soap:Envelope xmlns:soap=\"http://schemas.xmlsoap.org/soap/envelope/" 
               xmlns:xsi=\"http://www.w3.org/2001/XMLSchema-instance" 
               xmlns:xsd=\"http://www.w3.org/2001/XMLSchema\"&gt;
    &lt;soap:Body&gt;
        &lt;FUNC_NAME xmlns="http://tempurl.org"&gt;
            XML_PARAMETERS
        &lt;/FUNC_NAME&gt;
    &lt;/soap:Body&gt;
&lt;/soap:Envelope&gt;


Then in your code just read this file to string with File.ReadAllText and replace FUNC_NAME and XML_PARAMETERS with string.Replace with real values.