Following up on the answer by @MiffTheFox, you can use XPath together with LINQ to XML. Here's an example based on your sample data.
1) First, the namespaces you'll need:
using System.Linq;
using System.Xml;
using System.Xml.Linq;
using System.Xml.XPath;
2) Load your XML document into an XElement:
XElement rootElement = XElement.Parse(xml);
3) Define your XPath location path:
// For example, to locate the 'MimeType' element whose 'Extension'
// child element has the value '.aam':
//
// ./MimeType[Extension='.aam']
string extension = ".aam";
string locationPath = String.Format("./MimeType[Extension='{0}']", extension);
4) Pass the location path to XPathSelectElement() to select the element of interest:
XElement selectedElement = rootElement.XPathSelectElement(locationPath);
5) Finally, extract the MimeType value that is associated with the extension:
var mimeType = (string)selectedElement.Element("Value");