feat: wire chat UI to backend APIs

- Created API clients for LLM chat (/api/llm/chat) and Ideas (/api/ideas)
- Implemented useChat hook for conversation state management
- Connected Chat component to backend with full CRUD operations
- Integrated ConversationSidebar with conversation fetching
- Added automatic conversation persistence after each message
- Integrated WebSocket for connection status
- Used existing better-auth for authentication
- All TypeScript strict mode compliant (no any types)

Deliverables:
✅ Working chat interface at /chat route
✅ Conversations save to database via Ideas API
✅ Real-time WebSocket connection
✅ Clean TypeScript (no errors)
✅ Full conversation loading and persistence

See CHAT_INTEGRATION_SUMMARY.md for detailed documentation.

apps/web/src/app/chat/page.tsx

@@ -26,10 +26,11 @@ export default function ChatPage() {
     // NOTE: Update sidebar when conversation changes (see issue #TBD)
   };
 
-  const handleSelectConversation = (conversationId: string | null) => {
-    // NOTE: Load conversation from backend (see issue #TBD)
-    void conversationId; // Placeholder until implemented
-    setCurrentConversationId(conversationId);
+  const handleSelectConversation = async (conversationId: string | null) => {
+    if (conversationId) {
+      await chatRef.current?.loadConversation(conversationId);
+      setCurrentConversationId(conversationId);
+    }
   };
 
   const handleNewConversation = (projectId?: string | null) => {