You need to manually pull the files out of the computer. If you turn the computer around there are two screws on the back. Turn them left until they come out of the little holes they've been screwed into. Pull the panel off. Now you'll see a lot of wires and file folders. Pull the tray out containing all of the individual files look for one labeled 'Steam' in black marker. Once you have it, pull it out and put it next to you for now. While you are in there and the tray is out, if you think you don't have use for any other files you can pull them out and put them in a recycling container to free up some space. Now that you're done with that go ahead and push the tray closed and put the panel back on. Remember those two screws from earlier? Yep, grab them and push them back into the holes. This is tricky but you want to push them into the hole a little, apply pressure, and at the same time start turning them to the right. There are little threads on the screws that will "catch" and you'll just keep turning to the right until its tight and you can't turn anymore.
Congrats you did it! Now when the new computer comes in, do the same but instead of taking the folder out, you'll put it in.