webui: add Jump to Bottom button for mobile and desktop chat views

Add a floating Jump to Bottom button that appears when users scroll up
from the bottom of chat interfaces, positioned at the boundary between
chat content and input areas.

Co-Authored-By: sketch <hello@sketch.dev>
Change-ID: sba26f10bfcccf392k
diff --git a/webui/src/web-components/demo/mobile-chat.demo.html b/webui/src/web-components/demo/mobile-chat.demo.html
new file mode 100644
index 0000000..5fc1800
--- /dev/null
+++ b/webui/src/web-components/demo/mobile-chat.demo.html
@@ -0,0 +1,102 @@
+<!doctype html>
+<html lang="en">
+  <head>
+    <meta charset="UTF-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+    <title>Mobile Chat Demo</title>
+    <style>
+      body {
+        margin: 0;
+        padding: 0;
+        height: 100vh;
+        font-family:
+          -apple-system, BlinkMacSystemFont, "Segoe UI", "Roboto", sans-serif;
+      }
+    </style>
+    <script type="module">
+      import "../mobile-chat.js";
+
+      const chatElement = document.querySelector("mobile-chat");
+
+      // Simulate multiple messages to test scroll behavior
+      const messages = [
+        {
+          id: "1",
+          type: "user",
+          content: "Hello, I need help with setting up a project.",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "2",
+          type: "agent",
+          content:
+            "Hello! I'd be happy to help you set up a project. What kind of project are you working on?",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "3",
+          type: "user",
+          content:
+            "I'm trying to build a web application using React and TypeScript.",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "4",
+          type: "agent",
+          content:
+            "Great choice! React with TypeScript is a powerful combination. Let me help you get started. First, we'll need to set up the project structure. Here are the steps:\n\n1. Create a new React app with TypeScript template\n2. Install additional dependencies\n3. Set up the folder structure\n4. Configure the development environment\n\nWould you like me to walk you through each step?",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "5",
+          type: "user",
+          content:
+            "Yes, please walk me through each step. I want to make sure I understand everything.",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "6",
+          type: "agent",
+          content:
+            "Perfect! Let's start with step 1:\n\n**Step 1: Create a new React app with TypeScript template**\n\nOpen your terminal and run:\n```bash\nnpx create-react-app my-app --template typescript\ncd my-app\n```\n\nThis will create a new React application with TypeScript already configured. The `--template typescript` flag tells Create React App to set up TypeScript for you automatically.",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "7",
+          type: "user",
+          content:
+            "Okay, I ran that command and it created the project. What's next?",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "8",
+          type: "agent",
+          content:
+            "Excellent! Now let's move to step 2:\n\n**Step 2: Install additional dependencies**\n\nYou might want to install some commonly used packages. Here are some recommendations:\n\n```bash\n# For routing\nnpm install react-router-dom\nnpm install @types/react-router-dom\n\n# For styling (choose one)\nnpm install styled-components  # or\nnpm install @mui/material\n\n# For state management (if needed)\nnpm install redux @reduxjs/toolkit react-redux\n```\n\nWhich of these sound useful for your project?",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "9",
+          type: "user",
+          content:
+            "I think I'll need routing and probably Material-UI for styling. Should I install both?",
+          timestamp: new Date().toISOString(),
+        },
+        {
+          id: "10",
+          type: "agent",
+          content:
+            "Absolutely! React Router and Material-UI work great together. Run these commands:\n\n```bash\nnpm install react-router-dom @types/react-router-dom\nnpm install @mui/material @emotion/react @emotion/styled\nnpm install @mui/icons-material\n```\n\nMaterial-UI (now called MUI) requires @emotion packages for styling, and the icons package gives you access to thousands of Material Design icons.",
+          timestamp: new Date().toISOString(),
+        },
+      ];
+
+      setTimeout(() => {
+        chatElement.messages = messages;
+      }, 100);
+    </script>
+  </head>
+  <body>
+    <mobile-chat style="height: 100vh; display: block"></mobile-chat>
+  </body>
+</html>