fix(docs): Prevent "Ask Sui UI" button from disappearing after site navigation (#21940)

## How to Reproduce

- Open the main page: https://docs.sui.io
- Navigate to the "Guides" section by clicking the button in the navbar
(will get you to https://docs.sui.io/guides)
- The bug:
- On the live site, the "Ask Sui AI" button disappears until the page is
refreshed.
  - On this branch, the "Ask Sui AI" button stays visible as expected.

Author: ClockRide

docs/site/src/plugins/askcookbook/index.js

@@ -2,7 +2,8 @@
 // SPDX-License-Identifier: Apache-2.0
 
 // It is going to be exposed in HTTP requests anyway, so it's fine to just hardcode it here.
-const COOKBOOK_PUBLIC_API_KEY = "eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJzdWIiOiI2NjU5ODBiNDAwZTliZDQ2MzcwZDlhNzYiLCJpYXQiOjE3MTcxNDE2ODQsImV4cCI6MjAzMjcxNzY4NH0.0JCgi4bJ_f6ILFgEyYAP-KeCm1dzOKwH30tC3jEs2_A";
+const COOKBOOK_PUBLIC_API_KEY =
+  "eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJzdWIiOiI2NjU5ODBiNDAwZTliZDQ2MzcwZDlhNzYiLCJpYXQiOjE3MTcxNDE2ODQsImV4cCI6MjAzMjcxNzY4NH0.0JCgi4bJ_f6ILFgEyYAP-KeCm1dzOKwH30tC3jEs2_A";
 
 async function askCookbookPlugin() {
   return {
@@ -19,26 +20,33 @@ async function askCookbookPlugin() {
           },
           `
             <script>
-              function initCookbook() {
-                try {
-                  if (window.__cookbook_script) {
-                    window.__cookbook_script.remove();
-                  }
-                  const script = document.createElement('script');
-                  script.src = 'https://cdn.jsdelivr.net/npm/@cookbookdev/docsbot/dist/standalone/index.cjs.js';
-                  script.async = true;
-                  document.body.appendChild(script);
-                  window.__cookbook_script = script;
-                } catch (e) {
-                  console.error("Error while initializing Cookbook:", e);
+              window.initCookbook = function initCookbook() {
+                // It's a public API key, so it's safe to expose it here
+                const PUBLIC_API_KEY = "eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJzdWIiOiI2NzU4Y2YyODM4NTcxODU5MjU0MGIyMDciLCJpYXQiOjE3MzM4NzM0NDgsImV4cCI6MjA0OTQ0OTQ0OH0.Z3cv3HuMkYq3aYZYzCKkkYuM5LI3KG-kuA0R-GaSMV4";
+
+                let cookbookContainer = document.getElementById("__cookbook");
+                if (!cookbookContainer) {
+                  cookbookContainer = document.createElement("div");
+                  cookbookContainer.id = "__cookbook";
+                  cookbookContainer.dataset.apiKey = "${COOKBOOK_PUBLIC_API_KEY}";
+                  document.body.appendChild(cookbookContainer);
+                }
+
+                let cookbookScript = document.getElementById("__cookbook-script");
+                if (!cookbookScript) {
+                  cookbookScript = document.createElement("script");
+                  cookbookScript.id = "__cookbook-script";
+                  cookbookScript.src = "https://cdn.jsdelivr.net/npm/@cookbookdev/docsbot/dist/standalone/index.cjs.js";
+                  cookbookScript.async = true;
+                  document.head.appendChild(cookbookScript);
                 }
-              };
+              }
             </script>
           `,
         ],
       };
     },
   };
-};
+}
 
 module.exports = askCookbookPlugin;