How to build a 2 mile long runway on a 1 mile square island?
Anyone who has ever flown at Zar (EPZR) in Poland knows how much a runway slope can reduce the needed field length. For aircraft which need a 2 miles long level runway, a 1 mile long one with a 15% slope at its end will do just fine. Make the slope shallow at the bottom and increase it the further up the runway goes, just like the hillside runway at Zar. If the end point is 183.5 m higher than the touch-down point, the height difference is equivalent to a speed reduction from 60 m/s to zero.

Take-offs work in the opposite way, giving the same speed boost on the way down. A 1 mile long runway will do just fine. You do not even need to run it diagonally across the island.

Yes, this does not exactly answer the riddle, but is an engineering solution to a simple problem.