Burning smell in 2010 Corolla

are you mechanically in clined the brake pads might be worn and the rotors need to be turned or replaced. and have you looked under the hood to look at the belt

You may have got the cables hot that go to the starter. Continually trying to start the vehicle over a short period of time with no breaks in between can overheat the copper conductors. The conductors will get hot causing the insulation to heat up and if they get hot enough, it could potentially break down and melt/burn the insulation on the wire. This could explain the odor. I would recommend having the cables inspected for damage, especially on the ends of the cable where the connections are, as soon as you can. Have them replaced if the insulation, conductors, or connections have been compromised. This could be a potential hazard to the electrical system possibly causing shorting and resulting in larger problems.

Mark is right on. You should not crank it over for more than about 8 to 10 seconds at a time, and while starting do not touch the gas pedal. Wait at least 2 minutes before trying again, and do what Mark suggests.