FWIW after trying a ton of the suggestions in here, the one that just worked for me was to create a separate network name (SSID) within my router for the 5Ghz network. Once my two problem devices (iPad Air, iPhone 5S) were forced to connect to that 5Ghz-only WiFi SSID, BOOM, problem solved. Router is an older Airport Extreme (Fifth Generation).
My symptom had been as follows: iPad Air 2 and Android phone operating on WiFi at expected max speed given our Comcastic plan with an owned DOCSIS 3 cable modem (about 50MB down)... meantime my iPhone 5s and iPad Air painfully slow (like under 1MB down). Occasional bursts of speed, but not for long. This problem only cropped up after iOS 9, which aggravates me. My suspicion is that it's something in iOS 9 on these older devices that makes it lock on to the 2.4GHz band rather than automatically using the more-robust 5GHz band. I should also say, for anybody reading this, I live in a crowded urban environment with a TON of neighboring 2.4GHz networks stepping all over each other (if you're a PC user curious to see this what your area's WiFi networks look like, try out the free Xirrus Wifi Inspector, pretty interesting). I bet if I lived in the country I wouldn't have experienced this problem.
Anyway, hope this helps somebody out.