If you do have a backup of the My Endnote Library.DATA folder. Move the whole thing to a new place (not too deep in the directory structure. In particular I veer away from putting it in my documents, as that adds a whole nother long string of path to the name). As CrazyGecko mentions, the name as it stands is pretty unusual, and indicates some error that perhaps the whole of the documents & settings folder was copied to my documents at some point, becoming recursive).
But anyway. In the new .DATA folder location (NOT IN THE DATA FOLDER itself) create an empty text file with notepad with the name of the library (My Endnote Library.enl in your case, but I also recommend copying the library afterwards to a new unique name). Now try opening the newly created file and it should be recovered.