If you were to hard wire, meaning pluging in a networking cord, the HP and the router together, do you get the internet? So the HP was able to connect in the past? As of now, the only thing that I can recomend is to A: Check for a firmware update for the router.
1. Go to belkin
then click "Networking Products"
2. Select "Wireless", Then find your router in the list.
3. At the top of the FAQ section there will be something for, "How do I update my firmware" or something to that effect.
4. Follow the instructions, the download link is at the right of the page, towards the top, it says "Software & Updates", do the Firmware Update.
I am not familiar with Belkin routers, I use Linksys, but this should help.
Also, If you can get some sort of internet connection on the HP, do a windows update to see if it can find any.
1. In Internet Explorer, the web browser, go to Tools, then Windows Update.
2. Follow the instructions on screen to update your computer.
Hope this helps you both, if not, just post again, myself or somebody on this site will gladly give you more information.