This tip of Mike Snow shows you how to obtain your client’s IP address when using Silverlight.
Unfortunately there is no way to accomplish this directly from Silverlight. It is also not supported via JScript so communicating between Silverlight and JScript will also not help here.
The first way I have found to accomplish this is to have your Silverlight application communicate directly with a WCF service where the web services returns to the Silverlight client the IP address of the client.