Why is there so much overshoot in my LED circuit and how do I prevent it?
You have no capacitors across the power supply. That's going to cause problems, even with a good power supply.

Furthermore, "crappy long jumper wires" and "breadboard" throw stray inductance everywhere. You will see voltage overshoot whenever current abruptly changes according to \$v = L\:\mathrm di/\mathrm dt\$, where \$L\$ is some value determined by the crappiness of your long jumper wires, and the loops formed by your breadboard connections.

You also haven't given details of your measurement setup. If the ground lead of the scope is not very near the probe tip, this introduces even more inductance into your measurement device, and you will measure more overshoot than there is. At fast MOSFET switching timescales, even the inductance of the alligator-style clip typically provided with the probe can be significant.

I would hazard a guess that this problem isn't a problem, if you add some capacitors across the power supply, and build the circuit with a reasonable layout mindful of stray inductance.