I understand that you are having some issues in order to get the connection to your Slingbox since you have a new ISP (Internet Service Provider).
The fact that you connect the Slingbox to a new router means that you need to select the new IP address provided by the new router for your Slingbox. Once you connect the Slingbox to your new router, we recommend you to reset the router in order to pick up the new IP address. Then, you'll use the new IP address in order to create the rule opening the port 5001 through the router settings. If you do not know how to, just check the following links:
If you follow these indications and the issue persists, I recommend you to post back and make sure to include further details about your home network (make and model of your router, connections between the Slingbox and the router, etc). This way, other forum users will provide you with accurate responses.