Your problem *MAY* be the type of WEP key used:
64-bit WEP uses a key of 5 ASCII characters or 10 Hexadecimal characters.
128-bit WEP uses a key of 13 ASCII characters or 26 Hexadecimal characters.
However, some wireless hardware doesn't like an ASCII key, so normally a Hexadecimal key should be used. Your router may be configured with an ASCII key, but the new Laptop's adapter requires a 'Hex' key - you can tell which the existing key is by the number of characters, 5 or 13 is ASCII and 10 or 26 is Hexadecimal.
ASCII characters are basically all of the characters on your keyboard. The Hexadecimal characters are the numbers "0" to "9" and letters "a" to "f" only, a total of 16 available characters.
If you do have the wrong type of key, you need to go into the router's wireless security settings and set up a new WEP key. Having done so, you then need to remove the existing wireless "profile" in each of the computers and go through the procedure to "Connect" again when a new "profile" will be created - in other words, run the wireless adapter's software, scan for "Available Wireless Networks", find and highlight the one that matches your router's SSID (network name) and select the option to "Connect" to it. You should be prompted to enter any WEP or WPA encryption key and, once "Connected", the wireless adapter should get its IP address by DHCP from the router and you should have network and internet access.
To delete the profile:
In XP, if you're using Windows "Wireless Zero Configuration" to manage your wireless networking, go into the Network Connections folder ("Start > Run", type NCPA.CPL and click OK), right click the Wireless Network Connection, select Properties and then the Wireless Networks tab, where you can highlight and then "Remove" an existing profile.
If you're using third-party software provided with the wireless adapter, there should be a similar option to remove an existing profile.
In Vista, right click "Network" (My Network Places) and select Properties. Click the link at the left to "Manage Wireless Networks" then, in the next window, highlight the "Network" and click the Remove button