Get the model from the widget, instead of  the `documentManager`



### Problem
If the document is not saved, the cell sent to the LLM can be different from the one in the widget (as well as the whole Notebook, BTW), which can be very confusing.

### Proposed Solution

Get the model from the widget, instead of using the `documentManager`
### Additional context

Raised in https://github.com/jupyterlite/ai/pull/226#issuecomment-3522755759
