Apparently, this is a common fault in the otherwise fine Asus U46E-BAL5 laptop. A new display will cost some 350 USD (though the people at Asus say that the problem will likely pop up again, as the problem is on the engineering design). I decided to fix it myself.
(The photo you see here is a stock one taken from Asus.)
This is what I used:
* Several Hex Screws M2x10. (This means: 2mm diameter, 10mm long.)
* Several M2 Nylock Nuts
* Hex screwdriver
* Tweezers and pliers
* Rotary tool (B&D moto-tool) with accessories (2mm drill bits for metal, grinders, cutters, and the like)
* A piece of scrap metal (tin, aluminium)
* A bit of duct tape (à la MacGyver)
* Utility knife
Step 1: Remove Outer Plastic Shell to Reveal Hinges.
Use your hands for this (or the help of a plastic tool, e.g., a guitar pick).
Step 2: Remove the Broken Piece of Plastic (!!!) Attached to the Hinges.
Step 3: Drill Sundry Wholes Through the Screen Case
Take care not to touch the display or any electrical components.
Step 4: Attach Hex Screws With Nuts
Step 5: Attach Piece of Scrap Metal to Give Further Sturdiness
Step 6: Cut and Grind Protruding Screws
Step 7: Grind a Bit Outer Plastic Shell So As to Allow for the New Screws/nuts Inside