Some napkin math - total Internet bandwidth is estimated to be 3-4 petabits per second. Let's say 1% of that is inference traffic - 35 terabits - then that's ~4.375 trillion characters per second at 8 bits per character. A single keyboard press might require 2 mJ, so you need roughly 8.75 GW (electricity required for ~9 million homes) of continuous mechanical force to keep pace. This would cause the keyboard to heat up 175 million degrees C per second. Disappointingly, this is not enough to create a black hole but I'm sure Randall Munroe could tell you how the guy's pet T-rexes would be vaporized into previously unknown types of plasma and how satisfyingly whimsical that would be.

The keyboard springs are sooooo weak. It takes forever for them to recover after a press. Our dude would have to grab the key after each character and pull it back up so it can keep up with him.