Reliability problems with an array of 91 high-power LEDs

From the datasheet:

Figure 1. Absolute maximum forward current.

and further on:

Figure 2. Derating LED current at increased temperature.

Current and temperature are your problems. You are running at absolute max current with no wriggle-room and you are allowing the temperature to rise. At 60° ambient the max current allowed falls off dramatically.

You have already hit on the answer: your LEDs are getting hot. 15 watts may not sound like much, but it's building up and killing your LEDs. I suggest you get a thermistor and attach it to the center of the board, then monitor the temperature as the system operates. Even better, attach it to the body of one of the LEDs.

Because you're using this as a backlight, don't use narrow-beam LEDs. Use relatively wide beam units, and space them apart so air can flow through. If you can find a source of, let's say, 35 degree LEDs, install only every other one in a checkerboard pattern, soldering jumper as necessary. You'll only get half the total brightness, but that's barely perceptible, and the improved airflow should be a big help. You may also need to provide a fan with some ducting to keep the air flow through the array adequate.

And always include a temperature monitor. While not directly applicable, this YouTube video shows the principles of cooling. In your case, since you've got a forest of vertically standing LEDs, it is important not to let the LEDs touch each other, since this will block the flow of air.