feat(service-credential-tester): slice 1 — backend test endpoint + per-type routines

TestResult dataclass + translate_connection_error shared helper in base.py.
test_callable field on ServiceDefinition (default None). 7 per-type
test_connection routines (qbittorrent, prometheus via Grafana gateway,
alertmanager, jellyfin, authentik, ssh_tasks via build_ssh_client, nextcloud).
POST /api/services/test endpoint: validation-first (422 on malformed config),
dispatch, no-persistence, no-secret-logs. backups has test_callable=None.
qBit 'Fails.' → specific auth message (resolves #3 at API layer).
Backend: 362 pytest pass (+31 new), ruff clean. Frontend: build green.

backend/src/media_library_viewer_api/integrations/jellyfin.py

@@ -2,15 +2,40 @@
 
 from __future__ import annotations
 
+from typing import TYPE_CHECKING, Any
+
+from media_library_viewer_api.clients.jellyfin import JellyfinClient
 from media_library_viewer_api.integrations.base import (
     SecretField,
     ServiceBaseUrl,
     ServiceConfigBase,
     ServiceDefinition,
+    TestResult,
     WidgetConfigBase,
+    translate_connection_error,
     widget_kind,
 )
 
+if TYPE_CHECKING:
+    from media_library_viewer_api.services.settings_store import SettingsStore
+
+
+def test_connection(
+    config: dict[str, Any],
+    secrets: dict[str, str],
+    store: SettingsStore,
+) -> TestResult:
+    """Call JellyfinClient.users() — the lightest authenticated probe."""
+    try:
+        base_url = str(config.get("base_url") or "")
+        api_key = str(secrets.get("api_key") or "")
+        timeout = int(config.get("timeout_seconds") or 10)
+        client = JellyfinClient(base_url, api_key, timeout=timeout)
+        users = client.users()
+        return TestResult(ok=True, detail="Connected to Jellyfin.", evidence=f"{len(users)} users")
+    except Exception as exc:
+        return translate_connection_error(exc, context="Jellyfin")
+
 
 class JellyfinConfig(ServiceConfigBase):
     """Non-secret Jellyfin connection config.
@@ -68,4 +93,5 @@ DEFINITION = ServiceDefinition(
             refresh_interval_ms=30_000,
         ),
     ],
+    test_callable=test_connection,
 )