Writing from experience:
I didn't upgrade my mom's iPhone 6 from 10.3.3 for the longest time because I knew just how slow it would get, seeing my iPad just hit a wall after I did.
But one day it caught her by surprise and auto updated to iOS 11. Instantly the performance and battery was going haywire- one phone call and battery would go from 50% to 7%, lag, and generally be a liability to use.
Best option I had was to change the battery as it had some 868 cycles and showed battery health of 78% (shown under the iOS 11 battery health beta settings).
So got that done through the relatively inexpensive battery replacement program at Apple. Honestly, the best $30 I've ever spent cause the thing runs like a charm now! Got atleast another year left in it which is impressive for an almost 4 year old phone!
Hope this helps.
Tl;dr
Upgrade to iOS 11 and get the battery replaced through Apple's battery replacement program