You mean the battery got soldered on with the wrong polarity? Ouch, that can indeed smoke some components.
The schematic of the drone is here. From what I can see, with a bit of luck Q3 stopped the negative voltage from going into the rest of the system, but the motors are still directly connected to the battery. Given the symptom that one of the motors doesn't stop running, I'd expect the student fried Q4, Q5, Q6 and/or Q7. You probably also want to replace D1-D4; chances are high that those failed open because of the high current and even if the drone will work for a short while without them, they need to work otherwise those Qs will burn out again.