Well, I haven't finished the game yet, but 95% completed so far, so it is definitely possible with keyboard (azerty) only.
Most controls are fine, but Dash is difficult because with keyboard you get only 8 directions easily (and for diagonals you have to press 2 keys simultaneously, to get more precise angle you need to stop holding one of the key briefly, it is not simple).
I had hard time when the ginso tree was flooded, but then it became smooth again. Second really hard time was inside the volcano.
The main critic I have is that I often confuse keys x (fire) and c (dash) and ctrl (sprint) and shift (glide), because they are too close and I end up messing up a difficult run. I also use s (crash) scarcely because of the need to locate the s key under my fingers.
I am a casual gamer, it took me 25 hours and 1500 deaths to came to these 95%, so from what I seen in the walkthrough videos, it is smoother with pads, but certainly not impossible with keyboard only, I did it.