Let's try a couple things:
Go to the list of wireless networks, on the laptop.
You can then disconnect, if connected.
Then click the name that represents your network, and click CONNECT.
It will ask you one time for the access password. (the encryption key that you set, in the router, hopefully for WPA encryption)
Once you put the correct password in, it should remember it and work forever.
If you don't know the password, it's possible that the router isn't set up with a password. If so, you can go into the router. (example for Linksys brand, the address you would go to from a computer that WORKS is
http://192.168.1.1 admin/admin)
-Click Wireless Networks
-Click Wireless Security
-From there, you can set it to WPA encryption
-From there, you can also assign (or view) the 10 character password/key
-Save the changes
(Note the instructions above will vary somewhat, depending on the brand and model)
After that, you can go back to the wireless laptop. Disconnect and reconnect to your network, and supply the password/key entered above.
I hope this is helpful,
Mike