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/metrics/app.go

@@ -0,0 +1,53 @@
+package metrics
+
+import (
+	"github.com/prometheus/client_golang/prometheus"
+	"github.com/prometheus/client_golang/prometheus/promauto"
+)
+
+// MusicServer's own metrics. Labels only take values from small fixed sets: never song
+// names, IDs, tokens or IPs, since every value creates a new time series.
+var (
+	// SongsRequested counts random songs handed out, by how they were picked.
+	SongsRequested = promauto.NewCounterVec(prometheus.CounterOpts{
+		Name: "musicserver_songs_requested_total",
+		Help: "Random songs handed out, by how they were picked (random, low, classic).",
+	}, []string{"mode"})
+
+	// SongsPlayed counts songs marked as played, by how they were picked.
+	SongsPlayed = promauto.NewCounterVec(prometheus.CounterOpts{
+		Name: "musicserver_songs_played_total",
+		Help: "Songs marked as played, by how they were picked (random, low, classic).",
+	}, []string{"mode"})
+
+	// LibrarySoundtracks is the number of soundtracks in the library.
+	LibrarySoundtracks = promauto.NewGauge(prometheus.GaugeOpts{
+		Name: "musicserver_library_soundtracks",
+		Help: "Soundtracks in the library, counted at startup and after each sync.",
+	})
+
+	// LibrarySongs is the number of songs in the library.
+	LibrarySongs = promauto.NewGauge(prometheus.GaugeOpts{
+		Name: "musicserver_library_songs",
+		Help: "Songs in the library, counted at startup and after each sync.",
+	})
+
+	// SyncDuration is how long syncs take, by type.
+	SyncDuration = promauto.NewHistogramVec(prometheus.HistogramOpts{
+		Name:    "musicserver_sync_duration_seconds",
+		Help:    "How long a sync took, by type (full, changes).",
+		Buckets: []float64{1, 5, 10, 30, 60, 120, 300, 600, 1200, 1800, 3600},
+	}, []string{"type"})
+
+	// SyncLastSuccess is when the last sync without errors finished.
+	SyncLastSuccess = promauto.NewGauge(prometheus.GaugeOpts{
+		Name: "musicserver_sync_last_success_timestamp_seconds",
+		Help: "Unix time when the last sync without errors finished.",
+	})
+
+	// BrokenSongs is the number of songs the last full sync found missing and removed.
+	BrokenSongs = promauto.NewGauge(prometheus.GaugeOpts{
+		Name: "musicserver_broken_songs",
+		Help: "Songs whose file was missing in the last full sync. They are removed from the library.",
+	})
+)