I have not found a setting within the Steam mode to change input speed. I did boot into the desktop and slow down the mouse cursor there. But it does not seem to carry over into game mode.
The game I am playing does not have a cursor speed setting either.
I think mouse and keyboard only works properly in Desktop mode. Not sure if intended or a bug.
My suggestion is to open a support ticket with Valve/Steam. They might help you troubleshoot it, but most likely they will escalate the problem to their dev team as a feature request / bug report.