I had the exact same thing. Everything sluggish, delayed, like never before. It happened the second I downloaded IOS 11.2.1. I ended up calling support, they ran diagnostics, found nothing at first, but what I think it ended up being was I had LOTS of apps (even ones I haven't used in years) that I had not updated. So I "updated all" (which took a painfully long time) and then turned the phone off completely, and turned it back on. (Then I actually "opened" each app and close it to get rid of the blue dots). I've only been using it for a few hours, but I truly think this was the problem. Apple Support via chat (Rudy) advised that it is best to update all apps with a major change from one version like IOS 10 to another like IOS 11. Apparently app developers make lots of changes to optimize apps with a version change. Even things like messages, calculator and voicemail were very slow for me. But I think this process did the trick. I was concerned that it was related to all the battery publicity, but that didn't make sense that a low battery would affect performance.
iPhone 6s is running very slow after iOS 11 update
