feat: FastAPI app with REST API routes and WebSocket endpoint

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Co-Authored-By: Claude <noreply@anthropic.com>

tests/test_api_broker.py

@@ -0,0 +1,29 @@
+import pytest
+from httpx import AsyncClient, ASGITransport
+
+from mqtt_home.main import app
+
+
+@pytest.fixture
+async def client():
+    transport = ASGITransport(app=app)
+    async with AsyncClient(transport=transport, base_url="http://test") as ac:
+        yield ac
+
+
+async def test_broker_status_no_client(client):
+    resp = await client.get("/api/broker/status")
+    assert resp.status_code == 503
+
+
+async def test_broker_clients_no_client(client):
+    resp = await client.get("/api/broker/clients")
+    assert resp.status_code == 503
+
+
+async def test_health_endpoint(client):
+    resp = await client.get("/health")
+    assert resp.status_code == 200
+    data = resp.json()
+    assert "status" in data
+    assert "mqtt_connected" in data