Grumpy Mike was incorrect in post #3 about your LED's being AC, They are PWM'd internally.
Also incorrect (post #6) to suggest you buy an expensive AC volt meter for the sole purpose of measuring an LED ring current consumption to a hobbyist. (Again, not AC)
The suggestion of the Oscilloscope was correct. You can usually find them relatively cheap and functional on ebay for a used one, and even some of the USB based digital ones are affordable and some of those can even analyze data busses like I2C, SPI and oneWire on top of acting as a digital storage scope.
However, If you connect a large value capacitor from the LED side of your current measuring resistor to GND, it will smooth out the current through the resistor of the Pulsed Current enabling you to get an accurate reading even if your using an analog volt meter. See attached schematic, Test Point 1 is where you will connect your negative lead from your meter, the positive side of the meter goes to the other side of the resistor.
While not perfect, it should suffice to get a decent reading at least for a few of the leds on full bright. Also I'm still correct that the chart DOES prove Ohm's Law! The bigger the current draw, the bigger the voltage drop across the resistor, and the lower the voltage at the side of the resistor the LEDs are connected to.