You can add two new columns to your Kanban board called "Documentation" and "Documentation Review".
Then tasks cannot move to your "Done" column unless documentation is written and passes review. If you enforce column limits documentation it will also block other tasks if not completed.
In addition to this (and going a bit off topic). I've been adding checklists to Github PR templates (it's really easy[1]) for things like, "Did you re-read the relevant API docs? Do they need to be changed?" and it helps me a ton.
A separate openapi spec that is not enforced can quickly become outdated, then an auto-generated from code is better.