Measuring/confirming voltage and current
When you add your resistor and LED to a feeble 9 volt supply/battery, it's likely that the terminal voltage drops so, measure the voltage across the supply when you have your circuit (resistor and LED) connected.
So, you should probably trust the implied 8.13 volts under load. But, you have to remember that the multimeter measuring amps might add a few ohms of series resistance too.
So, with 5.8 volts across the resistor you should have a current of 58.23 mA but, when you insert the meter this is slightly less at 56.8 mA. The 56.8 mA would imply a resistance of more like 102.1 Ω. So, your meter's current shunt resistor is about 2.5 Ω.