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/lib/api/chat.ts

@@ -0,0 +1,57 @@
+/**
+ * Chat API client
+ * Handles LLM chat interactions via /api/llm/chat
+ */
+
+import { apiPost } from "./client";
+
+export interface ChatMessage {
+  role: "system" | "user" | "assistant";
+  content: string;
+}
+
+export interface ChatRequest {
+  model: string;
+  messages: ChatMessage[];
+  stream?: boolean;
+  temperature?: number;
+  maxTokens?: number;
+  systemPrompt?: string;
+}
+
+export interface ChatResponse {
+  model: string;
+  message: {
+    role: "assistant";
+    content: string;
+  };
+  done: boolean;
+  totalDuration?: number;
+  promptEvalCount?: number;
+  evalCount?: number;
+}
+
+/**
+ * Send a chat message to the LLM
+ */
+export async function sendChatMessage(request: ChatRequest): Promise<ChatResponse> {
+  return apiPost<ChatResponse>("/api/llm/chat", request);
+}
+
+/**
+ * Stream a chat message from the LLM (not implemented yet)
+ * TODO: Implement streaming support
+ */
+export function streamChatMessage(
+  request: ChatRequest,
+  onChunk: (chunk: string) => void,
+  onComplete: () => void,
+  onError: (error: Error) => void
+): void {
+  // Streaming implementation would go here
+  void request;
+  void onChunk;
+  void onComplete;
+  void onError;
+  throw new Error("Streaming not implemented yet");
+}