Have you attempted to uninstall and reinstall chrome? This usually fixes problems when something that should work does not. Seeing as accessing spotify works in this way works for me, it might just work for you.
If you click on Settings in Chrome, then scroll to bottom and click on Advanced Settings. you should see a Privacy section, click on the Contect Settings. This should open a new window. Scroll down until you see the Plugin section. Make sure the middle selection is ticked and click on Manage Exceptions. In the box type [*.]spotify.com and make sure allow is selected. Click done and that should be it fixed.