Your SlingLink Turbo acts as an ethernet bridge. They should be in fact transparent to the router. Only the devices that they interconnect should be visible one another. For example, if you have the router on one end of the SlingLinks and a Slingbox on the other then those devices will be able to communicate to one another.
I'm guessing your router isn't able to "see" the Slingbox because it has probably retained the network configuration of another router. I recommend performing a factory reset of your Slingbox (http://support.slingmedia.com/get/KB-1000038) to resolve it. This will return the Slingbox to the factory configuration which will allow it receive a new DHCP address from the router. Then you can go through the Setup Assistant to reconfigure the Video Source again.