Documentation: Screen reader fails announcing about the hint (Press Ctrl K) to invoke the search button.

## Description
User is not able to know the provided hint.
User with visual impairment who rely on screen reader will not be known about the short key to invoke the search button.

## Steps to reproduce

Open https://docusaurus.io/docs URL.
Navigate to search button and invoke it.
Listen to screen reader announcement.

**Live reproduction:**

https://codesandbox.io/s/github/algolia/docsearch/tree/main/examples/demo](https://docusaurus.io/docs)

## Expected behavior
Screen reader should announce hint (Press Ctrl K) ti invoke the search button.

## Environment

OS Ver: 25324.1011
Edge Ver: 112.0.1722.23 (Official build) dev (64-bit)
URL: [FAST](https://www.fast.design/)
AT: Screen reader.
DOC Search version: 3.1


https://github.com/algolia/docsearch/assets/93735775/6ac9df23-4e37-4775-b2c0-9627969b816e

https://github.com/algolia/docsearch/assets/93735775/070318a3-ab64-4b91-9d8a-2e39c344300e

Start with the live Docusaurus reproduction and inspect the search button behavior with a screen reader. Trace the TypeScript search implementation to determine how the Ctrl K hint is exposed; done means the screen reader announces the hint when the search button is focused or invoked.

Written by the indexing model from the issue text.