Hi Bára,
Thank you for posting, I am here to help. Possible reasons your Surface keyboard's backlight does not turn on when you start typing include misconfigured brightness settings, software issues, connection problems, and low battery.
Let us try this step to fix the problem:
To turn on the backlight of your keyboard, please make sure first that the Fn key is OFF (not lit up).
Kindly check then if the F7 key on your keyboard has a half sun icon or Prtscrn. If it is the sun icon, try to press it several times and check if the keyboard backlight lit up.
If the F7 key is for Prtscrn, please use then the F1 and F2 keys to adjust the keyboard backlight.
Let me know how this goes. Looking forward to helping you out. If you need more help please feel free to let me know. If the answer is helpful, please click "Accept Answer".
Best regards,
Jansher