How do I add a new node to a file with specific value from another file?

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C#

Hi, I'm trying to create a program having the following steps:
1) Get all xml files from a user given path
2) Open each of the files (if any) and search for nodes <institution> where it is in the format <funding-source><institution-wrap><institution>...</institution></institution-wrap></funding-source>
3) Get the value of the nodes <institution> and search the exact value in the database xml inside the nodes <skosxl:literalForm xml:lang="...">
4) If found, get the attribute value of its parent node <skos:Concept rdf:about="..."> minus the string http://dx.doi.org/
5) Add a node <institution-id institution-id-type="fundref"> in the xml file after the <institution> node with the value like <funding-source><institution-wrap><institution>...</institution><institution-id institution-id-type="fundref">VALUE of the rdf:about attribute</institution-id></institution-wrap></funding-source>
Here is a sample input file and the desired output for that file.

What I have tried:

string pathToUpdatedFile = @"D:\test\test2\Jobs\IEEE_sample - Copy.xml";
			var files=Directory.GetFiles(pathToUpdatedFile,"*.xml");
			foreach (var file in files)
			{
				var fundingDoc = XDocument.Load(@"D:\test\test2\Jobs\fundref.xml");
				XNamespace rdf=XNamespace.Get("http://www.w3.org/1999/02/22-rdf-syntax-ns#");
				XNamespace skosxl = XNamespace.Get("http://www.w3.org/2008/05/skos-xl#");
				XNamespace skos=XNamespace.Get("http://www.w3.org/2004/02/skos/core#");
				
				var targetAtt = fundingDoc.Descendants(skos+"Concept").Elements(skosxl+"prefLabel")
					.ToLookup(s => (string)s.Element(skosxl+"literalForm"), s => (string)s.Parent.Attribute(rdf+"about"));
				XDocument outDoc = XDocument.Load(pathToUpdatedFile,LoadOptions.PreserveWhitespace);
				foreach (var f in outDoc.Descendants("funding-source").Elements("institution-wrap"))
				{
					if (f.Element("institution-id") == null)
					{
						var name = (string)f.Element("institution");
						var x = targetAtt[name].FirstOrDefault(); // just take the first one
						if (x != null)
							
							f.Add(new XElement("institution-id", new XAttribute("institution-id-type","fundref"),x.Substring(@"http://dx.doi.org/".Length)));
					}
					outDoc.Save(file);
				}
				
				Console.ReadLine();

But it is not doing what I described above...can anyone help?
Also, there could be identical <skosxl:literalForm xml:lang="..."> values in the database xml file (but with different attrib (rdf:about) values), in that case take the value of the first match and write a txt file saying the value appears multiple times in the database or something like that just to notify the user if the correct value is inserted in the file.