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/MessageList.tsx

@@ -1,7 +1,7 @@
 "use client";
 
 import { useCallback, useState } from "react";
-import type { Message } from "./Chat";
+import type { Message } from "@/hooks/useChat";
 
 interface MessageListProps {
   messages: Message[];