For this call,
AFElement objElement = myDB.Elements["Name"];
"Name" is the name of the root element directly under the database. So if you want to find deeper down, you will need to do .Elements["AnotherName"] again. However, I would recommend you to use this call instead.
AFElement.FindElements and any of its overloads.
You mentioned that you have used this already. Could you try with null in the String parameter to see if any elements are returned?