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I am trying to set up my Arduino as a replacement remote control. The setup works and I can use it to turn on/off my things, for example my fan.

However, the working distance is quite short. Around a meter. The actual remote for my fan works across the distance of my apartment.

What can I do to improve the working distance of my Arduino remote? Does the LED need a back reflector or something? Does it need a different LED?

The LED I am using is this, it came in a bundle I purchased. https://www.amazon.com/dp/B07PJC7WSQ/#detail-bullets

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I assume that the led you purchased just isn't as bright as the led that came with your remote. As I've seen from your link you could quite easily swap the led from the board. If the new led has the same voltage, the board should work just fine with any other led. Maybe just search for a brighter led with the same wavelength.

I don't think that a back reflector would have such a big effect to boost the range to the distance you need. Also I think it would be rather complicated to install one on the tiny board.

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Judging form the picture of the IR breakout board you shown, it seems that the board is drive the LED directly via Arduino GPIO pin. The current of the GPIO pin could supply is typically around 20mA @5v (and about 10mA @3v3) which is not sufficient. A typical 940nm IR LED has the spec like this:

Max. Continuous forward current: 100mA
Max. Forward Voltage: 1.5 typ., 1.8V max.@50mA

In order to drive the LED, a circuitry like this would be required:

schematic

simulate this circuit – Schematic created using CircuitLab

Assuming the LED has a voltage drop of 1.8v, and you are supply the circuit with 3.3v, the 18 ohm resistor should limit the LED current to about 83 mA ((3.3 - 1.8)/18 = 83mA) - safely under its maximum rating of 100 mA. Use a 20 ohms if want to be on the safer side. The transistor's emitter must be connected to the Arduino ground.

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