[Docs] Implement search-index generation based on the `DocsBuilder`

[Luna-Klatzer]

Is there an existing issue for this?

• [X] I have searched the existing issues

Current State & Issue

Implement automatic search-index generation during the build of the static documentation and API reference files.

This should replace the existing manual search index.

Expected Changes & Improvements

• [ ] Implement automatic search index generation where a JSON file containing URL, title and description for every searchable page should be generated.
• [ ] Restructure and update the search bar code to use the new search index.

Referenced sites

• The entire website (https://kipper-lang.org)