So first things first, I had to disassemble the original GBA. It's just 6 tri-wing screws and one Phillips screw to open the case, and a further 3 Phillips screws to remove the mainboard.
Since the new case contains a full set of new screws and all the case parts, I've set the old ones aside as spares, and since the old screen works, it'll be a useful part to keep, as will all the buttons etc.
First thing I did was fit the screen. This is the easiest part, since it will only fit one way, and comes pre-installed with adhesive. I did a trial fit first as the display has some electronics on the left of the screen that have to slip under the edge, but once figured out, it was simply a matter of peeling the adhesive and dropping it into place.
Once in place there were two copper pads like the one shown here:

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And you can see in the following images that these are stuck to the front of the case, top and bottom, and allow for touch brightness control as they are capacitance sensing pads.
The display also has 3 optional wires that can be connected to allow the use of an on screen menu display to allow other settings to be changed, accessed by holding down R+L+Select. You can see them here:

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I've opted to install these wires and they connect as follows.
The select button connects to TP2 on the front of the PCB here just above the start and select buttons:

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Apologies for the splat of solder mask/conformal coating, but this was necessary due to battery corrosion on the board, and where a lot of the original coating had just fallen off.
The other 2 wires go to the R&L buttons respectively. I wrote R&L on the back of the display to make sure I understood the orientation with the GBA flipped on it's front.

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The 2 wires go to the terminals I've marked in blue on the switches. It should be noted that it is the left terminal of each switch. This is my next job and I'll update when it's done, as I'll be ready to test the display and the functions.
It should be noted that this kit contains a ribbon cable for both 32 pin & 40 pin GBA screens so it is a good option if you can't identify which yours has.