I was able to draw some traces to rebuild but could somebody say what's the element I marked as question mark? I think it should be some resistor, but I don't know its value.

R40 is 10k resistor (103 = 10 and 3 zeros = 10000). Most likely pull-up for the keyboard signals. Actually, I think the R40 is the one missing and the one that is sill there is R41 but the markings are now mostly gone so can't be sure. If so, both would be the same value I think - in that case just make sure the bottom side (as on the photo) of R41 is connected to 5V and if so just install some 4k7-47k resistor between 5V and L1 filter you've mapped. If that fixes the keyboard you can either put a proper SMD replacement in there, or (if the pads are gone) just leave that temporary resistor in - so long it doesn't touch and short anything it's good.

This mobo doesn't look half bad but make sure the vias around the battery spill are not corroded to the point where there isn't a connection between copper layers. Nothing can be done about the inner ones but that's mostly power planes and not super critital in that area, so you want to focus on the outside layers.

I just remade those broken traces, added the resistor and it partially worked: I received an error again, but I was able to hear the beep after clicking the key on the keyboard.
I checked all the connections and I have figured out that L1 element on the motherboard doesn't have one of the pads. I replaced it with random 68 uH inductor and now everything works fine.

@Deunan thanks for all the suggestions! Thanks to you, the motherboard works again.