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/components/chat/index.ts

@@ -10,8 +10,9 @@
  * ```
  */
 
-export { Chat, type ChatRef, type Message, type NewConversationData } from './Chat';
+export { Chat, type ChatRef, type NewConversationData } from './Chat';
 export { ChatInput } from './ChatInput';
 export { MessageList } from './MessageList';
 export { ConversationSidebar, type ConversationSidebarRef } from './ConversationSidebar';
 export { BackendStatusBanner } from './BackendStatusBanner';
+export type { Message } from '@/hooks/useChat';