Try going through the standard checks, in this order:
- Make sure there's at least 20% free storage
- Leave the phone on the charger (but running) an extra hour or more past 100%
- Kill all background Location and audio/music apps (including sneaky ones like Facebook)
- Turn off auto-update and any un-needed Notifications and apps in Background App Refresh Settings
- Put the phone in Airplane mode and reboot it (Slide to Power off, wait a couple minutes, then power on)
- Check performance of using apps (Camera, keyboard, etc.) before turning off Airplane mode and connecting to WiFi
If you turn off Airplane mode and the iPhone starts to slow down, something (Mail?, iCloud?, iMessage?) might be doing tons of downloads in the background. A friend of mine had like 10,000 unread emails and iMessages that were slowing down her iPhone while they kept trying to download. She did a bunch of mass deletes, and her slowdown problems gradually reduced.
Also, check Battery Usage. Whatever is draining the battery might also be contributing to slowing things down if left running.
Please post results here if any of this helps fix the slow downs.