There were lots of problems with the original circuit. You've made some changes, but it's still a bit wonky.
Assuming your LED is that 3mm green LED that appears to be on the breadboard, your current should not exceed about 15-20mA tops. The forward voltage will be around 2.5V.
The LM358 op-amp has an output that can swing down to 0V on a 5V supply but cannot go higher than a few volts. Since Vbe is 0.7V or so we should limit the voltage across the sense resistor R1 to something reasonable, say 0.5V. So R1 = 0.5V/0.02A = 25 ohms. That is chosen so that there's enough voltage for the LED but the voltage is much higher than the few mV offset of the LM358.
Now divide your 3.3V maximum Vref down to 0.5V with something like 10K/1.8K and apply that to the non-inverting input. The 1.8K to GND will also deal with the input bias current if you disconnect the input.
The compliance of the resulting current sink at 20mA out is about 5V - 0.1V - 0.5V = 4.4V so the 2.5V LED has plenty of margin.
Maximum dissipation of the transistor into a short is 4.5V * 0.02A = 90mW which is fine. Leaving the LED open will cause the op-amp to attempt to drive 20mA into the transistor base, which it can do without damage at room temperature.
This particular circuit will likely work without the cap/R2 but the pair improves stability.
Unless you connected power to the LM358 backwards and it got hot, it's unlikely you have damaged it, they're pretty rugged devices. However, your measurements do not look good. Check the wiring first, and try another op-amp.