Limited normally means you aren't pulling internet but you have an IP address handed out by the router.
Since you connected the xbox, I'm guessing it's not a wifi password issue.
You might try giving the laptop a static IP and see if that fixes the issue.
http://ask-leo.com/how_do_i_configure_my_static_ip_address.html
You mirror all settings on another working computer, excpet give it a different number on the end. Choose something like 10 more than how many devices connect to your router. Say you have 4 devices, and this is the 5th one. Give the laptop x.x.x.14 Usually the pool is 2-50 so that should work.
What's the xbox's distance from the router? Is there any thing that might block the signal in between the router and xbox?