Hi
Sorry but you cannot get a connection through another computer if the computer is Off.
I am not sure that I totally understand the topology. However, if the Desktop is connected with cable to a Router for Internet connection, you can solve the problem by buying a simple Network switch.
You take the cable out of the Desktop and plug it to the switch. then you plug both the desktop and the Laptop to the switch.
That what network switch does, it splits one cable to be able to feed many computers.
Example of a Switch that can be connected with one cable to a Router and feed 5 computers.
http://www.newegg.com/Product/Product.aspx?Item=N82E16833166034-
Jack - Microsoft MVP, Windows Networking. WWW.EZLAN.NET