Hi all,
I've had this problem since 1,5 years now and I found a solution. I already ordered a new flex cable on ebay, but that didn't work. This is only the case when the backlight comes on but the phone doen't show anything on the screen.
Yesterday I tried to fix a ylod on my playstiation 3 and I had to reball the graphics card with a heat gun. After that was fixed, I thought: why not try this on my Nexus 5?
So I took out the motherboard and first warmed up the whole board slightly with the heat gun set to low to prevent localized thermal expansion. The board should feel warm but not too hot.
Now place the motherboard on a flat surface (like an old book or something) and place the heat gun about 5 cm above the ribbon cable connector and heat it on low setting for about 20 seconds. Watch out that the plastic ring in the vibration motor hole doesn't start to burn (as it did in my case). When you're done, make sure you DON'T TOUCH the board until it it's completely cooled down. Place it back in your phone. Make sure you place all the connectors back (I forgot the battery as it slipped under the board) and plug in power.
My phone is working like it never stopped!