|just thought i would leave the numbers incomplete as not sure they are supposed to be made public?|
It's just your external (public) IP you need to worry about. Your LAN IP is likely a private class C address in the 192.168.0.0/24 range.
Do you have a SOHO Router in your LAN that sits between the internet and your home network? If so, your default gateway address and DNS address should be the IP address of the LAN side of your router.
If you're able to get proper connectivity while connected to the wired interface of your laptop, compare the output of the ipconfig command to when you try it wirelessly and look for differences.
Oh to save you time putting information from a command prompt window up here, there's a quick and dirty method to copy/paste it from there to here and for info on how to copy/paste from a command prompt window click on my name above and read my "how-to" guide titled, "Post the output of ipconfig (version 2)"
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.