If your router is 802.11g wireless router, then disable the 802.11b wireless adapter and only use that 802.11g adapter.

From the network output you gave, your adapter didn't receive the correct IP, netmask, gateway and DNS form router. You need to confirm there is proper DHCP configuration with enough IP to assign on the router.  Use http web access to access your router's management page to confirm it. Read your router manual to help you. If you don't remember the router admin username and passwd, you can press the reset button to reset it (don't do this unless you need it, you need to reconfigure the router after resetting it). After that you can try again.

The other way to solve your problem is manually assign the IP, netmask and other network info on this problem notebook. Can give this a try. Just type the following command in command prompt (start -> run -> type cmd.exe)  on the other 2 working  computers.

ipconfig /all

Take down the IP, netmask, gateway and DNS info. Let say those 2 computers use 192.168.1.100 and 192.168.1.101 IP addresses, then you assign the 192.168.1.102 to your computer wireless's adapter. The netmask, gateway and DNS are the same as other computers. After that type ipconfig /all in command prompt to make sure those network info are assigned correctly, then try to ping the gateway IP and other computers.

Additional info to help you in case you need it:

Wireless Router Configuration

Wireles Adapter Configuration in XP

Wireless Adapter Configuration in Vista

Hope helps..

Wink
picoHat