Here in the video you can see the problem. I'm not alone in this as a quick google search will show. The laptop was basically tearing itself apart every time I opened or closed the lid.
I decided to fix this in an over-the-top steampunk kind of a way. I bought some half inch brass strip from ebay. The screws on a laptop are 2mm so I got some M2 machine screws too. From trial and error I ended up with one 25mm long and one 20mm long. (I know, mixing my units, terrible!) That and a hammer, screwdriver, hand drill and a 2mm bit and I was all set.
The photo shows me very carefully removing all twelve screws from the back. If you're being professional (I wasn't) you'll get a bit of paper or a photocopy of the back so you can put your screws back where they came from!
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Step 1: Let's Void That Warranty!
So, I removed the whole back (with the laptop powered down and battery out) and had a good check that there was nothing between the top of the case and the screw holes. When I found out it was clear, I replaced the back but left the two screws out that were each side of the big crack in the laptop.
Then I drilled through from the back to make a hole come out on the front. I used a hand drill and a fresh 2mm bit because I didn't want to do any more damage than necessary.
For bonus marks, you can see a previous repair with epoxy where this problem means that the power socket has been replaced already.
Step 2: Trial Fit, Mark, Drill and Repeat
I drilled a hole in one end of the brass strip, then loosely held it in place with a bolt. I bent the brass strip to roughly fit the laptop. I then tapped another bolt through the next hole to make a mark which showed me where to drill the next hole. I carried on doing this all the way around the laptop.
Step 3: Cut to Length and Bolt It Up
Once I had four holes, I cut the brass strip to length and bolted it through, making sure to tighten everything down.
As you can see from the final video, the movement has reduced by around 90%. If I were to do it again, I'd maybe mock it up in cardboard and use that as a mask for drilling and pay more attention to the bends around the contours of the case. And maybe glue it down as well.