Fair question. I think one of the things that feels good about jenga i general, and your game in particular, is the "physicality" of it... the very act of moving the blocks just feels satisfying. I'd want to recreate the feeling of direct manipulation of the physical game as far as possible using the mouse, which probably implies dragging the blocks rather than controlling them with WASD. I can see the problem, but maybe constraining the blocks to only moving in the horizontal plane (same as with WASD) and then mapping the mouse pointer onto that could work? It would also leave the non-mouse hand to do camera panning and tilting during movement operations, which could come in handy. Or maybe WS could be used for vertical movement and AD for rotation around the up/down axis, to be able to perform a clean and careful extraction? I think this would need to be prototyped.
In general I believe that the feel of manipulation is probably the most important thing to get right with this kind of game.