Did you already configure your router to use WEP?
The airport will not prompt you for WEP/WPA/WPA2 password unless you've actually specified that you want to use it on the router itself.
Who set up your wireless router? Get them to help you. If it was you, then make sure the Router is using WEP. If it is then you maybe try putting in the password with a '$' as the first character. Either you're a non-english speaker or are just rushing but you describe the problem poorly. You shouldn't have to turn off another computer to get WEP to work. They're totally unrelated.
Excuse my ignorance...but what's the $ for when entering the WEP key?