12V 10A linear LED supply keeps failing; need a better design
A simple fix for you: Don't put all your LED's in parallel. The reason you heat up so much is the high current and low voltage. The high current heats up your resistors and MOSFET, and both the low voltage and high current heat up your linear regulator. 

You state in your instructable that your LEDs are burning 2W each. If the 5A current is evenly divided amongst the seven LEDs, that's 0.7A and 2.8V each.

Try stacking 6 LEDs (Using six because of the even number and the chance you don't have eight) in two parallel branches of three. For the same brightness and LED wattage, you now only need to push 1.4 A (0.7 x 2), and the voltage across the three LEDs in series will be approximately 8.4V (2.8 x 3). Two parallel branches of four would be fine as well, just don't let the required voltage exceed what the regulator can drive (Dropout voltage and current resistor voltage drop).

In this new configuration, the resistor heating will be only 30% of previous, MOSFET heating will be only 8% of previous, and the current regulator heating will be only 11% of previous!! Another possible benefit: You will pull far less electrical power from your aircraft (~30% of previous).

Calculations:
Old:
MOSFET heating: \$I^{2} \times R_{DSon} = P \rightarrow (5A)^{2} \times 0.5\Omega=12.5W\$
Resistor heating: \$I \times V = P \rightarrow 5A \times 1.25V = 6.3W\$
Regulator heating: \$I (V_{in} -V_{out})=P\rightarrow 5A*(13V - (2.8_{V_{LED}} + 1.25_{ V_{res}} + 2.5_{ V_{MOS}}) ) = 41.3W\$  

New:
MOSFET heating: \$(1.4A)^{2} \times 0.5\Omega=1.0W\$
Resistor heating: \$1.4A \times 1.25V = 1.8W\$
Regulator heating: \$1.4A*(13V - (8.4_{V_{LED}} + 1.25_{ V_{res}} + 0.7_{ V_{MOS}}) ) = 3.7W\$   

Note: LED brightness balancing can be tricky. Better balance would have each parallel string of LEDs with its own current regulator, but this is mostly cosmetic. Having all the LEDs on the same heatsink as you do will help with keeping the brightness even. Here is a decent discussion of different LEDs series and parallel layouts. An extra consideration is reliability due to the aerospace application. If one LED fails to an open circuit, the whole series path will go dark. The current source will try and push twice the current through the other line, which might just brighten it, or might be enough to burn it out. Having two regulators with their own series LEDs creates a completely redundant segment, like having two separate lightbulbs. 

Further Note: Paralleling other components is an easy way to increase the current capacity of other sections. It does come with a few considerations, but can help you reach much higher amperages with the same components. See this page for linear regulators, and this page for MOSFETs.