Hi.
The left over part is the small black ground clip on the upper screw of the battery connector.
The picture show you where to put it :)
[image|467262|align=center]
You can see it on step 5 in the "[guide|7111|iPhone 4S Battery Replacement]"-guide
Just slide it underneath the batteryclip
For the error 4013: Put the left over part back in right place ;)
Check the dock connector. Is it connected to the logic board correct?
If all seems fine, you could try to restore thru DFU-mode. Check this guide: http://thebigboss.org/guides-iphone-ipod-ipad/dfu-mode