Seems like you have a short circuit somewhere.
NB The LED will only come on if you tell thr pin it is attached to go low. Its not a power-on indicator. Sort out your short circult first as that may cause the LED not to light up anyway.
Since you've using breadboard - turn the power off - and remove some of the unneeded components - such as the LED and its resistor, the servos, the ISP etc - so just about everything except the power supply and the chip. Turn the power back on and check that the regulator doesn't get hot. Once you've sorted out the problem then add some of the other bits, one at a time, (with the power off) and keep checking for heat. Until you get the lot back on the board.