feat: Workspace Proxy picker show latency to each proxy (#7486)

* chore: Add cors to workspace proxies to allow for latency checks
* Add latency check to wsproxy

Use performance API timings.
- Fix cors and timing headers
- Accept custom headers

coderd/coderd.go

@@ -806,6 +806,17 @@ func New(options *Options) *API {
 		return []string{}
 	})
 	r.NotFound(cspMW(compressHandler(http.HandlerFunc(api.siteHandler.ServeHTTP))).ServeHTTP)
+
+	// This must be before all middleware to improve the response time.
+	// So make a new router, and mount the old one as the root.
+	rootRouter := chi.NewRouter()
+	// This is the only route we add before all the middleware.
+	// We want to time the latency of the request, so any middleware will
+	// interfere with that timing.
+	rootRouter.Get("/latency-check", LatencyCheck(api.AccessURL))
+	rootRouter.Mount("/", r)
+	api.RootHandler = rootRouter
+
 	return api
 }