Perhaps consider that you cannot interact, by touch, with your external display. As such, when using the iPadOS16 Stage Manager feature, you’ll not be able to navigate or control App windows on your external monitor using touch.
If instead you are mirroring your iPad screen to the external monitor - in lieu of using Stage Manager’s windowed environment - the iPad’s onscreen keyboard is accessible.
A limitation of the current Stage Manager implementation is that the iPad’s own “onscreen” touch-UI can only be used with the iPad itself - and cannot interact with App windows displayed on an external monitor. As a consequence you’ll need to use either an external TouchPad, or Mouse; these interfaces can seamlessly interact with - and control - Apps and App windows displayed on both the iPad and external monitor.
As for the keyboard, the current Stage Manager UI appears to presume that an external keyboard will be used alongside the Keyboard/Mouse. While it is possible to use the onscreen iPad keyboard in limited circumstances, providing keyboard input to Apps displayed on an external monitor is difficult.