How can I use the stock Android settings app on an LG G2?
Short answer: You can, but you shouldn't. Install a custom ROM instead.

Long answer: Some features in your stock ROM may be dependant on that ROM's Settings app, because it might contain some preferences that affect them. Your Settings app may be missing some Lollipop settings simply because the ROM doesn't support them. You can try to install the AOSP Settings app, but there's no guarantee it will fully work because of those two issues. A solution to that would be to flash an AOSP (or AOSP-like) ROM, such as CyanogenMod, probably the most popular one out there.