I don't have the PC version, but did you try to see if the 2nd player is mapped on the same keyboard? I'm an old timer, PC games back in my day used to do that.
go to the controller settings. it shows you what the layout is for 2 people playing on one keyboard. also, go buy a usb gamepad they are dirt cheap. or a psx to USB adapter, also dirt cheap.
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I am going to have 1 of everything. someday...