I’m unable to replicate your issue - albeit with another iPad model (with iPadOS 15.6 installed) that supports split-keyboard. When using the native Messages App the split keyboard, in either Landscape or Portrait screen orientation, appears fully below the messages window.
Perhaps try a forced-restart of your iPad - as so doing will often resolve an isolated software issue:
- On an iPad with a Home button: Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.
- On other iPad models: Press and quickly release the volume up button, press and quickly release the volume down button, then press and hold the top button. When the Apple logo appears, release the button.
If you continue to experience difficulties, you may need to seek assistance from Apple Support. You can contact the Support Team using the Get Support link at top-right of this page. Alternatively, you can initiate and manage your support cases using the excellent Apple Support App. If not already installed on your iPad, the App can be downloaded from the App Store:
https://apps.apple.com/gb/app/apple-support/id1130498044
You might also visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to offer assistance:
Genius Bar Reservation and Apple Support Options - Apple