Add Prometheus metrics endpoint and instrumentation (#21)

* Add Prometheus metrics endpoint and request instrumentation

Expose operational metrics at GET /metrics (unauthenticated) using the
metrics + metrics-exporter-prometheus crates. Instruments HTTP request
counts/durations, active/streaming connections, cache hits/misses,
signing durations, upstream latency, and error counters across all
proxy modules. Existing GET /v1/metrics (backend passthrough) is
unchanged.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* Address Copilot review feedback on metrics PR

- cache.rs: Add comment that moka entry_count() is O(1)
- signing.rs: Split signature duration histogram by algorithm for
  per-algo performance visibility
- metrics_middleware.rs: Improve panic message in setup_metrics_recorder
  to help operators debug recorder conflicts
- error.rs: Reorder upstream error branch for clarity (comment before
  counter, no functional change)

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.6 <noreply@anthropic.com>

Author: bowenwang1996

Cargo.lock

@@ -43,6 +43,8 @@ thiserror = "2"
 anyhow = "1"
 tracing = "0.1"
 tracing-subscriber = { version = "0.3", features = ["env-filter"] }
+metrics = "0.24"
+metrics-exporter-prometheus = "0.16"
 
 [dev-dependencies]
 wiremock = "0.6"

Cargo.toml

@@ -28,11 +28,19 @@ impl ChatCache {
     pub fn set_chat(&self, chat_id: &str, value: &str) {
         let key = self.make_key(chat_id);
         self.inner.insert(key, value.to_string());
+        // moka::sync::Cache::entry_count() is O(1) (atomic counter), safe to call per insert
+        metrics::gauge!("cache_size").set(self.inner.entry_count() as f64);
     }
 
     pub fn get_chat(&self, chat_id: &str) -> Option<String> {
         let key = self.make_key(chat_id);
-        self.inner.get(&key)
+        let result = self.inner.get(&key);
+        if result.is_some() {
+            metrics::counter!("cache_hits_total").increment(1);
+        } else {
+            metrics::counter!("cache_misses_total").increment(1);
+        }
+        result
     }
 }