My guess is that the lens element that is right behind the eyepiece has come loose. This element moves back and forth when you turn the diopter adjustment wheel at the upper right side of the eyepiece.
Use the guide you linked in your question to get the top cover off. Once you've done this, you can use this guide to get an idea of the necessary procedure to disassemble the viewfinder. This guide is for the 550D, the preceding model, so the procedure might not be exact but it will probably be close enough.
I haven't found a proper service manual for the 600D, but there is a parts manual which has exploded views and might be useful. Get it here. You'll want to look at pages 5 an 10 specifically. The drawings are true vector line drawings so you can enlarge the !&&* out of them, which is way cool.
When dealing with the optics, wear latex gloves to avoid leaving fingerprints. Especially the focusing screen on the bottom of the viewfinder assembly is extremely delicate and must be treated with all due respect.
Once you have the eyepiece off, you'll most likely be face to face with the offending lens element. Given that there probably is not enough room for the lens to have flipped itself around its orientation will most likely be correct. It might just have popped out of its frame, but it might also be that the frame is broken. In the former case, a bit of patience might be sufficient to pop the lens back in place. In the latter case, you have a problem because you wouldn't want to use e.g. a cyanoacrylate glue on optics (cyanoacrylate tends to fog optics).
If you're going to take the plunge, take pictures if you can and let us know how you get by.