Google Account Manager 5.1.1 ← I don't have the "Google Login Service" APK, but I think it's an older version of this; try his version, then this if it doesn't work
Reboot, then install:
If you're having issues, you might want to Settings → Applications & Parental Controls → Manage applications, then go to each Google App listed above, in reverse order, and Clear Data, then Clear Cache, then Uninstall. When done. reboot and start over - it's possible older versions are causing issues.
Personally, I installed Home Switch, as Jason recommended, and use it with the Google Now Launcher (which will likely make you update some other things first), but... I think you're just as well off, if not better, by just sticking to the Fire OS.