Normally, people replace both Lcd (to display pictures) and sensor screen ( to receive touch signals). If you are sure which part is broken, then you can replace it individually. In your case, I suggest you replace both of them. You can buy just Lcd.
Hi Tom! Supposing that all of the new parts (logic boards, display, lvds cable) are in good condition and you assemble them well to the motherboard, I think this problem is caused by hardware, the motherboard (main board). I suggest you should check as following to be sure 1. Open your MacBook in a dark room (in the evening without lights around, for example). 2. Look closely to the screen (display) of the laptop. - If you can see dim pictures (though it's hard to see), then the power supply from the mainboard to the screen fails - If you can see weak light, then video signals from video chip (GPU) fails. Check the GPU chip, it process and give out LVDS (Low Voltage Different Signal, digital video signal) to the display. Hope these tiny suggestion can help!