Expose Prometheus metrics on a separate port

Metrics are served on METRICS_PORT (default 2112, "off" to disable), never on
the public port. Besides the Go runtime and process metrics this adds:

- http_requests_total and http_request_duration_seconds per route pattern,
  from an Echo middleware.
- musicserver_songs_requested_total and musicserver_songs_played_total by
  pick mode (random, low, classic). The mode is now stored with each queued
  song (migration 000006, not released yet).
- Library size, sync duration, last successful sync, broken songs from the
  last full sync, and the database connection pool.

Refs #35, #36, #37

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

internal/backend/sync.go

@@ -19,6 +19,7 @@ import (
 
 	"music-server/internal/db/repository"
 	"music-server/internal/logging"
+	"music-server/internal/metrics"
 
 	"github.com/MShekow/directory-checksum/directory_checksum"
 	"github.com/google/uuid"
@@ -199,6 +200,7 @@ func (s *SyncService) run(ctx context.Context, full bool) {
 	run := &syncRun{q: s.q, musicPath: s.musicPath, full: full, progress: func() { s.foldersSynced.Add(1) }}
 	result := run.execute(ctx, func(total int) { s.foldersTotal.Store(int64(total)) })
 	result.TotalTime = formatDuration(time.Since(start))
+	s.recordMetrics(ctx, full, time.Since(start), len(result.CatchedErrors) == 0)
 
 	if s.onComplete != nil {
 		s.onComplete(ctx)
@@ -218,6 +220,42 @@ func (s *SyncService) run(ctx context.Context, full bool) {
 		zap.Int("errors", len(result.CatchedErrors)))
 }
 
+func init() {
+	// Start both sync types at 0, so the series exist before the first sync.
+	metrics.SyncDuration.WithLabelValues(syncType(false))
+	metrics.SyncDuration.WithLabelValues(syncType(true))
+}
+
+// syncType is the type label of the sync metrics.
+func syncType(full bool) string {
+	if full {
+		return "full"
+	}
+	return "changes"
+}
+
+// recordMetrics updates the sync and library metrics after a sync.
+func (s *SyncService) recordMetrics(ctx context.Context, full bool, took time.Duration, success bool) {
+	metrics.SyncDuration.WithLabelValues(syncType(full)).Observe(took.Seconds())
+	if success {
+		metrics.SyncLastSuccess.SetToCurrentTime()
+	}
+	if err := s.UpdateLibraryMetrics(ctx); err != nil {
+		logging.GetLogger().Error("Failed to count the library", zap.Error(err))
+	}
+}
+
+// UpdateLibraryMetrics sets the library size metrics from the database.
+func (s *SyncService) UpdateLibraryMetrics(ctx context.Context) error {
+	count, err := s.q.CountLibrary(ctx)
+	if err != nil {
+		return err
+	}
+	metrics.LibrarySoundtracks.Set(float64(count.Soundtracks))
+	metrics.LibrarySongs.Set(float64(count.Songs))
+	return nil
+}
+
 func (r *syncRun) execute(ctx context.Context, setTotal func(int)) SyncResponse {
 	var err error
 	r.existing, err = r.q.GetAllSoundtracksIncludingDeleted(ctx)
@@ -623,6 +661,7 @@ func (r *syncRun) removeBrokenSongs(ctx context.Context) {
 	}
 	_ = g.Wait()
 
+	metrics.BrokenSongs.Set(float64(len(broken)))
 	if len(broken) > 0 {
 		if err := r.q.RemoveBrokenSongs(ctx, broken); err != nil {
 			r.addError("RemoveBrokenSongs", err, "")