You can export the private key with the command-line tool from GPG. It works on the Windows-shell. Use the following command:
gpg --export-secret-keys
A normal export with --export will not include any private keys, therefore you have to use --export-secret-keys.
Edit:
To sum up the information given in my comments, this is the command that allows you to export a specific key with the ID 1234ABCD to the file secret.asc:
gpg --export-secret-keys --armor 1234ABCD > secret.asc
You can find the ID that you need using the following command. The ID is the second part of the second column:
gpg --list-keys
To Export just 1 specific secret key instead of all of them:
gpg --export-secret-keys keyIDNumber > exportedKeyFilename.asc
keyIDNumber is the number of the key id for the desired key you are trying to export.