How can kinetic energy be proportional to the square of velocity, when
velocity is relative?
Without reading the rest of your question, I must first reply that one has nothing to do with the other.
Kinetic energy is frame dependent, just as velocity is.
Momentum is proportional to velocity and is frame dependent too, just as velocity is.
Now, looking at the body of your question:
Imagine you and your mate are in space drifting along together, at an
unknown speed.
Unknown speed relative to what? Unknown speed relative to Earth? Unknown speed relative to the solar system? Unknown speed relative to the CMB?
Assuming 100kg spaceships, will 5kJ of energy always produce 10m/s of
relative velocity?
Relative to what? Relative to the initial inertial frame of reference before the acceleration? Or relative to some frame of reference in some arbitrary relative motion?
(The point of all these questions is to prompt you to think more clearly about your question in the hope that you'll come to the answer yourself...)
It is zero, because kinetic energy is the one associated only with an object in motion. It is maximum just before it touches the ground. And once it reaches the ground, most of the energy is perceived as sound, lost as heat and some as stress, which causes deformation of the body. If the body can't sustain the stress on its impact with the ground, it breaks.
Best Answer
The gravitational potential energy released during your fall through the atmosphere shows up as an increase in your velocity and hence your kinetic energy as you accelerate downwards. As your speed through the air increases, so does the viscuous drag being applied to your body by the air through which you are moving.
Terminal velocity is reached when the rate of release of gravitational energy during your fall is exactly balanced by the rate of energy loss because of air friction. Ignoring for now the increase in air density with decreasing altitude, at that point your velocity and hence your kinetic energy remains constant for the remainder of your descent and all the "extra" gravitational potential energy being released beyond that point is being continuously dissipated by air friction.
That dissipated energy shows up as sound waves (noise) and heat in the air.