Changed theme but googlebot still sees old, infected pages? How?

One of my clients today discovered that his Google search results contain lots of Viagra ads. So I found your Anti-Malware plugin and registered (and paid since it has so many positive recommendations). It found a file in my active theme that definitely looks like a backdoor for uploading files.

I also found several unusual files in the WordPress root containing long strings of bad-looking stuff so I deleted those files.

Those things didn’t help so I disabled all plugins and changed to the Twenty Fifteen and Twenty Fourteen themes. Now when I look with a normal browser I see my site with the desired theme. BUT when I use a browser configured to send the Googlebot user agent I STILL GET THE original theme with viagra links all over it.

Any advice about where to look, or what to do?