So you're seeing about 1.5v on the output, but the LM324N datasheet says that the output can drive down to at least 20 mV (into a 10K load). It also seems unlikely that something is pulling the output high(er). This would imply that the problem is on the input side of things.
The datasheets says that the inputs can go to 0V, but I suspect that this is not entirely true. It might be that they only go to "almost" 0v. Even a couple of mV higher than 0v might be enough to mess up your circuit.
I suggest changing your circuit. It's a rather major change, unfortunately. The easiest (from a testing standpoint) would be to switch to a different opamp that can run off of a +/- power rail. If the opamp has a negative rail then it will be fine if the input is close to 0v (and not V-). You might be able to run the existing opamp that way, but the datasheets are ambiguous about that (and I'm too tired to look into it closely).
A different change, which would be much more complex, unfortunately, would be to change the circuit so that the input to the opamp never gets close to 0v. There are several ways to do this, but none of them are easy. One way is to put the current sense resistors on the high side, between the power rail and your LED's. Then use a resistor-divider to bring that voltage down within the range of the opamp. This might require changing some polarities and such (swapping the + and - inputs), but don't take my word for that-- actually figure it out first.
Another possibility where this circuit is going wrong is that it is actually unstable. You might need a cap between the opamp output and the negative input, and a resistor between the transistors and the negative input. I usually figure that stuff out in a simulator, but if you wanted to guess at some values I would go with 10K resistor and a 100 pF cap.
You didn't supply a link to the datasheet, so this is only a guess.
Check the common mode input range of the opamp and see if it includes 600 mV. If not, then what you see should be no surprise.
Check the supply voltage range. Can it really handle 24 V.
Another possibility is that this particular opamp is broken. Probably not, but trying a second one is a good idea if you've checked the above two points and everything is within spec.
You should also have a bypass cap across the opamp power pins.
Best Answer
In theory your calculations are absolutly right and your results would make sense - IF the OpAmps could acutally output these voltages.
I can only assume that the question is targeting some thinking about plausibility: The upper OpAmp is supplied only by +/- 12V - so it will never be able to pull the ouput as low as -65V. It will go into saturation around -12V (let's assume the OpAmp can drive the ouput right to the supply rails. Most real OpAmps would need some headroom and would not even be able to go that low).
The same is true for the lower OpAmp: For the output to go to 8V, the OpAmp would need a positive supply rail of at least 8V, but it only has 6V available. So the theoretically absolute maximum output voltage will be 6V (again, assuming the OpAmp is capable of driving the output right up to the supply rail).
This would give output voltages of -12V and 6V.
But even these values seem a little bit unreasonable, when we take the resistor values into account: With V01 at -12V and the negative input at +12V this results in 24V/44 Ohm = 545mA of current that the upper OpAmp has to sink just from the input current. With V02 at 6V, there will be an additional current of 6V-(-12V)/3 Ohm = 6A! So the upper OpAmp would have to sink 6.5A. The lower OpAmp would have to deliver this 6A + 6V/6 Ohm (for the lower current path through the 2x 3 Ohm).
Without any detailed question I don't know what exactly they want as an answer, but the calculations show, that KCL will fail in this case, because the OpAmps go totally into saturation.