You don't need to have Linux already installed on the drive in the Toshiba Protege in order to use the procedure given in the SL Wiki article, but you do need a CD-ROM drive at least, as you would use a LiveCD to perform the procedure. Does your Protege have a CD-ROM drive? If not, then, providing the 2.5" HDD you mention is compatible with both the Protege and the laptop with DVD drive, then you could try to do what you asked, i.e. temporarily install the spare 2.5" HDD in the other laptop and use a LiveCD/DVD to install Linux on it and then follow the procedure in the SL Wiki.
However, as the Protege has USB ports, an alternative to the above would be for you to install a compact Linux such as Puppy Linux or Damn Small Linux on a pen drive (Google throws up lots of pages with instructions on how to do this from eiher Linux (HDD or LiveCD/DVD) or Windows -- here are just a couple: Installing to a USB Flash Drive
, Puppy Linux on USB flash Pen drive
) and use that to format and partition the drive in the Protege and then download the SL ISO directly onto the Protege via the Web and follow the instructions in the SL Wiki article to set up the Protege to boot from the ISO. Note that the gparted application is used to partition and format drives.
EDIT: The following article may also help you if you have Windows installed on the HDD of the Protege:Install any Linux distro directly from hard disk without burning any DVD
There are a lot of pages on the Web about doing this kind of thing.