docs(api): restore generated ChatKit API index (#3705)

- [x] I understand that this repository is auto-generated and my pull
request may not be merged

## Changes being requested

Restore the existing ChatKit section in `api.md` from the
[recorded public generated
snapshot](https://github.com/openai/openai-python/blob/0d3e70da47bb645fbfd0dd16fdd37ea10ee981f5/api.md#L633).
The handwritten legacy Realtime section had displaced it. This change
retains
that section and every other existing line, including documentation for
handwritten public helpers.

Only `api.md` changes: 50 lines are restored verbatim. All documented
ChatKit
types and methods already exist in the public SDK. No runtime code,
tests,
dependencies, API reference, compiler, or generation metadata changes.

## Additional context & links

Validation:

- Exact-byte comparison with the public generated ChatKit section.
- All eight restored source links and both import examples resolve to
existing
public files/exports; all six documented methods exist in both sync and
async
  resources.
- The repository's `scripts/utils/ruffen-docs.py api.md` formatter and
  `git diff --check` pass.
- The verified public custom-code report keeps 40 mixed files, with only
`api.md` changed and 39 customizations untouched. Its patch shrinks from
**+87/-32 to +105/-0**: the API index no longer deletes generated
content.

No tests are removed or relocated in this documentation-only cleanup.