For Android 5.1.1 (API 21), you can typically use versions marked for "Android 5.0+".
By downloading the latest Google Play Store APK compatible with Android 5.1.1, you bypass the broken auto-update queue. You get: google play store apk download for android 51 1 better
Open your current Play Store, go to – this forces a manual check for updates (often fails on Lollipop, but try anyway). For Android 5