Electronic – Why does the intensity of light in LEDs not increase with current after a particular value

The resistor can be on either side of the LED, but it must be present. When two or more components are in series, the current will be the same through all of them, and so it doesn't matter which order they are in. I think the way to read "the resistor must be connected to the anode" as "the resistor cannot be omitted from the circuit."

I needed to find the Light output power (LOP) and divide it with the input power to give the EQE, as EQE = Output Power / Input Power.

This is not the usual definition of quantum efficiency. Quantum efficiency should be a ratio of numbers of quanta. For an LED this would be $$\frac{\mathrm{Photons\ emitted}}{\mathrm{charge\ carriers\ injected}}$$

If you have the input current and output power you can calculate this as

$$\frac{P_o/h\nu}{I/e}$$

where \$P_o\$ is optical output power, \$h\nu\$ is the photon energy at the emission frequency \$\nu\$, \$I\$ is the injection current, and \$e\$ is the electron charge.

You do not need the diode forward voltage to calculate the quantum efficiency. You might need it in the future if you are asked to calculate the power efficiency or wall-plug efficiency.

How am I supposed to find the total Light output power like I have with the data in spreadsheet 1 then?

The ususal way to get the total light output of an LED is to put the LED at the input port of an integrating sphere, and measure the optical power at the output port with a large-area photodiode. Then scale the measured power by the loss factor of the integrating sphere.

Or am I calculating everything wrong and the light output power is given in another way?

It looks to me like you did not collect enough data when you were doing the experiment. You need to re-do the experiment and measure the light output for all of the input current levels you are interested in.

Realistically, there's no reason the quantum efficiency (my definition, not yours) should change dramatically at current levels below 1 mA. Maybe your next experiment will be with a laser instead of an LED, then you will see some interesting behavior changes at low currents.