Curved runways have many issues. Especially in bad weather, aircraft sometimes run off of straight runways, so a curved one would be even harder to stay on. The curve would also increase the distance needed for both takeoff and landing, meaning that even more than 2 miles would be needed.
Sloped runways do exist. However, sloped runways like EPZR (mentioned by Peter Kämpf) or Lukla tend to be small (both are less than 2000 feet, EPZR is not paved). The physics limit the runway to operating in one direction only. Though wind becomes less of a factor, the issues still prevent this from being used for large airports.
Another issue is the height of the high end of the runway. Zar and Lukla are both located in terrain that requires one end of the runway to be raised. A 1-mile square island would probably not be shaped this way. Even if it is, if an airplane goes off the runway, or needs to abort a takeoff or landing, the slope will make this much harder. The pilots would also have to adjust their takeoff and landing technique to account for the slope, which will probably result in many more hard landings and tail strikes. A sloped runway may be acceptable for small aircraft and short runways, but typical aircraft needing 2 miles of runway will be much more difficult to operate this way.
Some aircraft carriers solve the runway length issue with a slope only one the very end. However, this is most helpful to aircraft like the Harrier that already can provide some vertical thrust, and the aircraft must still perform an arrested landing. Other carriers use a steam catapult to assist takeoff, which requires much more maintenance but is capable of launching heavier aircraft.
Using a sea plane to land on the water is another option. However, aircraft on the scale of needing a 2 mile runway are typically not sea planes. If you are going to design a new plane, you might as well design one that can operate to the runway length requirement.
Another fun option is jet-assisted takeoff (JATO). Rocket engines are strapped to the plane to accelerate it to takeoff speed more quickly. These can be used for landing as well, but that is a bit more difficult.
Do to the issues with these options, large airports in need of land tend to just make more land. You can put an entire airport with runways >2 miles long on a completely man-made island. There is typically enough room around an island that creating an extension is not an issue.