Display faded to black and standby light blinks 14 times. Sony support said it is an error code and I am trying to find out what the error code is.

These blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices.
http://electronicshelponline.blogspot.com/
If blinks continue without any pause, the main board is faulty.
Search the site by type in the "brand name" to your device, in the "Search box at the top right of the display window of the Homepage" to get gathered related posts.

It is unusual. Count the blinking codes once again.
These blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices.
http://electronicshelponline.blogspot.com/
If blinks continue without any pause, the main board is faulty.
Search the site by type in the "brand name" to your device, in the "Search box at the top right of the display window of the Homepage" to get gathered related posts.

These blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices.
http://electronicshelponline.blogspot.com/
If blinks continue without any pause, the main board is faulty.
Search the site by type in the "brand name" to your device, in the "Search box at the top right of the display window of the Homepage" to get gathered related posts.

These blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices.
http://electronicshelponline.blogspot.com/
http://electro-medical.blogspot.com/
Blinking codes details of many popular brand TVs are given. Surf the site with patience.

14 times blinks is a rare case to Sony Bravia sets. Just count it one more time.
These blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices. Viewing it in "Mosaic" or "Magazine" will make surf easy.
http://electronicshelponline.blogspot.com/
Blinking codes details to many popular brand TVs are given. Surf the site with patience.

You have a couple of problems if you are sure there are 2 different blink codes. 13 blinks indicates a backlight balancer error which would explain your no light condition. 14 blinks would be a TCON board error which is the driver board for the LCD screen. Either error would keep the screen black.

Just count the number of blinks occur at one time. 14 blinks to trinitron TVs are not possible. It might be 10. If so,

1. Check Q8014 and Q8013 for shorts, leakage, or bad solder connections on the " D" board. If the transistors are bad, check R8051 and IC8005, also on the "D" board. If you wish to know more about blinking codes to Sony Trinitron models TVs, check this link. If it is Intermittent problems? This is a great feature. You can also bring up these fault codes on-screen to see a list of the problem numbers with the abbreviation of the problem and the number of times this problem has occurred. This is slightly different than entering the service mode. To see the diagnostic screen, press the following buttons: Display, Channel, 5, volume Minus, ( not plus ) then Power. This brings up a screen with a list of the problems and number of times they have occurred. Each of the problems named, are abbreviated as, "OCP", "OVP", "V STOP", ETC. We have repeated them in parentheses in the text for the "Blinks" described above so you can correlate between the diagnostic screen and the blink codes.

   
   
   

14 blinks to Sony Brand LCDs indicates a 'Backend error'. It means that, the main board to this TV should be checked for fault. In most cases the main board might need replacement, and a software update too. If you wish to know more details about this blinking code, check this link. It has details of blinking codes to several brands of TVs including Sony. Actually the work is a skilled one, need experience.

Since the TV is still receiving power, the problem is going to be with the TV's mainboard.  After you unplugged the TV and plugged it back in, you said the power light went out again, so I assume the standby light wasn't flashing again?  The flashing standby light is indicating an error code so my suggestion is to contact Sony for service options regardless if there is a warranty or not because service is the only solution to this situation.  Don't forget to provide them the error code or 4 flashes repeating.

Those blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices.
http://homenol.blogspot.com/
http://electronicshelponline.blogspot.com/
If blinks continue without any pause, the main board is faulty.
Search the site by type in the "brand name" to your device, [Exclude the model number] in the "Search box at the top right of the display window of the Homepage" to get gathered related posts.