fix: enhance type hints for health check and root endpoints

src/datacenter_docs/chat/main.py

@@ -4,7 +4,9 @@ Chat server stub for development.
 TODO: Implement full chat server with WebSocket support.
 Currently this is a minimal stub to allow the development environment to start.
 """
+
 import logging
+
 from fastapi import FastAPI
 from fastapi.middleware.cors import CORSMiddleware
 
@@ -30,13 +32,13 @@ app.add_middleware(
 
 
 @app.get("/health")
-async def health_check():
+async def health_check() -> dict[str, str | bool]:
     """Health check endpoint."""
     return {"status": "ok", "service": "chat", "implemented": False}
 
 
 @app.get("/")
-async def root():
+async def root() -> dict[str, str]:
     """Root endpoint."""
     return {
         "message": "Chat server stub - not yet implemented",