The 169.254.x.x is an APIPA address that is automatically assigned when DHCP fails to deliver an IP address.
This would indicate that currently the internet does not work on either the Laptop nor the Desktop.
Rather than using connection sharing, you would be better off bridging the two adapters together.
Connect like so:
[Router / Wifi]--------[Laptop]--------[Desktop]
Turn off the connection sharing. You should have internet access on the laptop at this point, or you have other issues.
Go in to "Change Adapter Settings" and Ctrl-click both the Wireless adapter and wired adapter so that they are both selected. Right-click and select
This will cause your Desktop computer to connect "directly" to the wireless network. The only other limitation here might be specific to your wireless hardware. The router or your wireless adapter may limit what MAC addresses are able to communicate over the connection, meaning it will block access to the desktop.
Make sure there are no 3rd party firewalls that might be getting in the way of this. I can only imagine what one of those will do in this setup.
Also, this is pretty much a non-issue nowadays, but technically you are supposed to be using a Crossover Cable to connect two devices of the same type together. Most modern network adapters automatically detect this, and adjust, so this will likely not be a problem.