LibWeb: Use the proper in-flight request to check if a stream is closing

Author: https://github.com/trflynn89
Commit: https://github.com/SerenityOS/serenity/commit/b6501adef8
Pull-request: https://github.com/SerenityOS/serenity/pull/23795
Reviewed-by: https://github.com/kalenikaliaksandr
Reviewed-by: https://github.com/kennethmyhra ✅

Tests/LibWeb/Text/input/Streams/WritableStream-write.html

@@ -0,0 +1,28 @@
+<script src="../include.js"></script>
+<script>
+    asyncTest(done => {
+        const stream = new WritableStream({
+            write(chunk) {
+                return new Promise(resolve => {
+                    println(chunk);
+                    resolve();
+                });
+            },
+        });
+
+        function sendMessage(message) {
+            const writer = stream.getWriter();
+
+            for (const chunk of message) {
+                writer.ready.then(() => writer.write(chunk));
+            }
+
+            writer.ready.then(() => {
+                writer.close();
+                done();
+            });
+        }
+
+        sendMessage("Well-hello-friends!");
+    });
+</script>