I think we can try to build further on Joel’s answer that one can “enumerate the positions for which white or black can force a draw by means of forcing the position into a closed finite space of positions. But this is not the same as forcing a draw, since perhaps black can force the play to continue indefinitely, without forcing it into a finite closed space of positions.”
There are two cases in which black can force a draw: forcing the position into a closed finite space of positions and forcing the play to continue indefinitely. Case 1 is decidable.
Consider case 2. How would a situation be like if black can force the play to continue indefinitely? First, white can be left with insufficient material to checkmate.
Secondly, the situation where a white mating formation (a few knights) is unable to decrease the distance between itself and the black king. We know that a single knight moves at a speed twice the speed of a king. The formation of two knights move at a speed equal to the speed of a king. Thus, on an empty part of the board black king is able to keep a constant distance between itself and a two-knight formation. Moreover, it is well know that two knights alone cannot force a checkmate (in fact even 2 knights with a king on a finite board cannot). It is a case of insufficient material. Thus, we know that a possible white mating formation has to consist of at least 3 knights and it implies that such a formation is slower than the black king. Black can force a draw if the black king can go to an empty part of the board.
Now, it seems that the only problem which remains is whether case 2 always has to take a form of insufficient white material or the black king being able to go to an empty part of the board.