OK, I'm confused - the "wireless dongle" simply receives the signal from your router, no? How exactly it would change your IP address???
If you have a dynamic IP address, then you would need to disconnect and connect again - login to your router (e.g. 192.168.1.1) and disconnect; one of my connections has a dynamic IP and I do that all the time, sometimes it takes 2-3 disconnects to change the IP
of course, you can change your IP by restarting the router, but it takes much longer (a few minutes in my case) while disconnecting and re-connecting from the router admin takes a few seconds