Maybe. It is real hard to say something can't be done with computers, but I know that I have the same issue with HP products when the client can't find their disc or the hard drive restore can not be done because the drive completely dies. I always have to grab the OEM version to reinstall and use their Product Key. This also happens in organizations that have volume licensing version. You have to use the volume licensing disc with the volume licensing key.
Glad you got it working. Just a head's up though... forum users are not allowed to talk about any methods that are not acceptable legal practices. In this case, since you have your COA, legally you are 'likely' ok.
On another note, the Dell Installation CDs that don't require a product key have a key preprogrammed into the install. Dell must have licensed the right to do so from Microsoft. I have never seen it done with any other manufacturer.