You'll need a flashlight to shine though the back side of the lid (Apple Logo) are you able to see the faint images of your icon's on your desktop? If you do then only the backlight is gone which also explains why the diagnostics didn't see it as a fault.
You'll need to test the logic board, do you have the needed tools and skill's?
=== Update (05/01/2017) ===
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@p OK, no need to go further.
+
@pierrickp - OK, no need to go further.
You've isolated the fault down to the displays ribbon cable and/or the logic board connector as the most likely fault.
Follow this IFIXIT guide: [guide|12341|MacBook Air 11" Mid 2012 Display Assembly Replacement]. Start off doing the first three steps (1 - 3), then jump down to Step19. Inspect the internal displays connection do you see any corrosion or is the connection not fully secured? Disconnect and carefully lift the cable does the cable look damaged in any way?
If you see something take a picture so we can see and post it in a fresh answer.
You'll need a flashlight to shine though the back side of the lid (Apple Logo) are you able to see the faint images of your icon's on your desktop? If you do then only the backlight is gone which also explains why the diagnostics didn't see it as a fault.
You'll need to test the logic board, do you have the needed tools and skill's?
=== Update (05/01/2017) ===
-
OK, no need to go further.
+
@p OK, no need to go further.
You've isolated the fault down to the displays ribbon cable and/or the logic board connector as the most likely fault.
-
Follow this IFIXIT guide: [https://www.ifixit.com/Guide/MacBook+Air+11-Inch+Mid+2012+Display+Assembly+Replacement/12341|MacBook Air 11" Mid 2012 Display Assembly Replacement]. Start off doing the first three steps (1 - 3), then jump down to Step19. Inspect the internal displays connection do you see any corrosion or is the connection not fully secured? Disconnect and carefully lift the cable does the cable look damaged in any way?
+
Follow this IFIXIT guide: [guide|12341|MacBook Air 11" Mid 2012 Display Assembly Replacement]. Start off doing the first three steps (1 - 3), then jump down to Step19. Inspect the internal displays connection do you see any corrosion or is the connection not fully secured? Disconnect and carefully lift the cable does the cable look damaged in any way?
If you see something take a picture so we can see and post it in a fresh answer.
You'll need a flashlight to shine though the back side of the lid (Apple Logo) are you able to see the faint images of your icon's on your desktop? If you do then only the backlight is gone which also explains why the diagnostics didn't see it as a fault.
You'll need to test the logic board, do you have the needed tools and skill's?
+
+
=== Update (05/01/2017) ===
+
+
OK, no need to go further.
+
+
You've isolated the fault down to the displays ribbon cable and/or the logic board connector as the most likely fault.
+
+
Follow this IFIXIT guide: [https://www.ifixit.com/Guide/MacBook+Air+11-Inch+Mid+2012+Display+Assembly+Replacement/12341|MacBook Air 11" Mid 2012 Display Assembly Replacement]. Start off doing the first three steps (1 - 3), then jump down to Step19. Inspect the internal displays connection do you see any corrosion or is the connection not fully secured? Disconnect and carefully lift the cable does the cable look damaged in any way?
+
+
If you see something take a picture so we can see and post it in a fresh answer.
You'll need a flashlight to shine though the back side of the lid (Apple Logo) are you able to see the faint images of your icon's on your desktop? If you do then only the backlight is gone which also explains why the diagnostics didn't see it as a fault.
You'll need to test the logic board, do you have the needed tools and skill's?