It seems that you have some issues in order to get the remote connection to your Slingbox ever since you installed a new home router.
Whenever you install a new router, it will provide a new IP address for all the connected devices. This means that even though the Slingbox is working fine at home, the IP address provided by the new router is different that the one provided by the old router.
With this in mind, you need to reset the Slingbox and access your new router interface in order to look for the new IP address. Once you get it, you will create a new rule on it. Finally, you'll need to use the same IP address in order to run the Setup Assistant for Internet Viewing. This way, the Slingbox will communicate with the Internet service through the port 5001.
The following links will provide you with further details and steps to follow.