I understand that you are having some issues with your Slingbox Classic ever since you replaced your old router.
The router is the device that provides the IP address every network device needs in order to reach the Internet. Every time you connect the Slingbox to a router, it will modify its settings according to the new router configuration. This means that the Slingbox has a new IP address and you must set it up again for Internet Viewing.
The following article will provide you with all the information about this specific situation and the steps to follow in order to fix this issue.