When the NSP is split the output is not in NSP format or in any format, just "File".
Do I just rename it with a ".nsp" to make them NSPs?
No, you don't touch these files. The script should have created a folder for you that ends with .nsp (and these files should be inside that folder). You just enable archiving on that specific folder and Tinfoil will recognize it as a nsp file.