Thank you both for your answers.
I’ve read in different places that some of the MX models should have an extra button to turn it on but sadly this one doesn’t seem to have any other than the main button. I’ve also tried pressing the button gently but it doesn’t even feel like the button was designed for being pressed in different manners.
So I suppose I’m left with having to buy a replacement remote which don’t seem to be awefully expensive anyway.
Otherwise, could there be a way to make this work with an Arduino and a IR led? Or do you think I will face the same issue with the frequency being too high for what the Arduino can emit?