Presumably the router is a new model. Most will still support WEP but WPA is simpler and safer.
Start by reading the quick-start leaflet that came with the router (or downloading the manual -- either from the CD which came from it or from the manufacturer's site).
Connect computer to router by ethernet cable (it makes life simpler when setting up). Open your browser and in the address bar put the IP address of the router (usually 192.168.0.1 or 192.168.1.1 -- but see manual). You should be greeted by a login box -- User Name is usually admin and the Password is usually password (but see manual if not).
Once in go to Wireless Settings and choose WPA-PSK TKIP as your security protocol. Choose a passphrase. Save setting. Exit.
Apply the same choices to your wireless adapter's utilities (or Windows Networking). Disconnect ethernet cable and see if you can connect wirelessly.
You should move the computer further away from the router -- such close proximity, supposedly can cause problems. Personally I rather doubt this is the cause of your problem, but it is received wisdom not to run them too close.
I would test the Asus with a different wireless router and see what happens.
You have, of course, turned the Asus' wireless on - it's sometimes a switch on the front or side of the laptop case -- or can be keyboard combination.
Did you test the Asus with the new router ?
If changing the wireless adapter in the Asus also didn't solve the problem it may be something you're overlooking.
I can only suggest going over the instructions in the manual again. WPA should not be an issue with recent equipment.