Add the Chrome Playpen Dashboard URL to the manifest file of the Chrome Web Store App.

chrome/browser/resources/webstore_app/manifest.json

Index: chrome/browser/resources/webstore_app/manifest.json
diff --git a/chrome/browser/resources/webstore_app/manifest.json b/chrome/browser/resources/webstore_app/manifest.json
index 598455a84eb45858f91cde8cc201a1ab0dfad354..7a520a400d1d9f354aaa0f387abaf3b5216dddd2 100644
--- a/chrome/browser/resources/webstore_app/manifest.json
+++ b/chrome/browser/resources/webstore_app/manifest.json
@@ -1,7 +1,7 @@
 {
   "key": "MIGfMA0GCSqGSIb3DQEBAQUAA4GNADCBiQKBgQCtl3tO0osjuzRsf6xtD2SKxPlTfuoy7AWoObysitBPvH5fE1NaAA1/2JkPWkVDhdLBWLaIBPYeXbzlHp3y4Vv/4XG+aN5qFE3z+1RU/NqkzVYHtIpVScf3DjTYtKVL66mzVGijSoAIwbFCC3LpGdaoe6Q1rSRDp76wR6jjFzsYwQIDAQAB",
   "name": "Web Store",
-  "version": "0.2",
+  "version": "0.3",
   "description": "Chrome Web Store",
   "icons": {
     "16": "webstore_icon_16.png",
@@ -12,7 +12,8 @@
       "web_url": "https://chrome.google.com/webstore"
     },
     "urls": [
-      "https://chrome.google.com/webstore"
+      "https://chrome.google.com/webstore",
+      "https://chrome.google.com/manage"

[not at google - send to devlin, 2015/07/29 17:48:47]

I didn't realise this is what we agreed on...

It's still not quite right, the extent of the webstore does not include
"manage", this is basically a workaround for a principled solution[*].

A side effect will be that on ChromeOS if you have a tab to
chrome.google.com/manage open, clicking on the webstore app icon will refocus
that window.

As I also mentioned, the management console doesn't need access to all the
webstorePrivate APIs, only that 1. It would be better to pull that out
separately.

However, fixing it properly will be time consuming. Please reference that in the
bug that you will file on this.

[*] in which we define a new API, and instead of having an app, we simply expose
that API to chrome.google.com. There is a bit of extra work here.

     ]
   },
   "permissions": [