Can you go at a coffee shop or something and try?
I am thinking the problem is your router. Router configuration issue might be the problem, and malware for routers is not unheard off. In fact, security of home routers isn't great. Just recently manufactures started to pay attention, and that is after numerous security breach. Now I am not saying you have a malware on your router, it could be simply a configuration problem, but we can't exclude the possibility, hence why I suggested you go somewhere else for your wireless-goodness for ruling out the Surface Pro 2 problem.
OH, just remembered. Make sure your date and time is correct.
And be sure to click on "Check for update", and not "Retry".
If you installed any security software (beside what is included in Windows 8), please uninstall it, it might be blocking Windows update.
Another thing you try, and change your DNS server from your ISP, to Google's.
Google DNS are: 8.8.8.8 and 8.8.4.4.
For more info:
https://developers.google.com/speed/public-dns/