From what I read you are asking how to get 8.1 on a USB am I right ?
They have updated the link and you just pick the language, windows 8.1 and x64 if it is x64
Then while in the setup it should ask you like at a normal install for the key and from what I know 8.1 or 8.0 the key will be accepted.
http://windows.microsoft.com/en-US/windows-8/create-reset-refresh-mediaPS: If your laptop was out of the box with Windows you don't need to make a fresh install just do a recovery using the recovery partition. I think you can upgrade to 8.1 from the store afterwards.
Good luck, hope that helped and that I understood what your problem was ^^