Why won't iOS 8 download on my iPhone?

For the past few days I've been trying to download the new operating software (iOS 8.0) on my iPhone 5, but I keep getting an error message message. When I try to download the software, it says "Downloading", followed right away by "Update Requested... Your software update request has been received. You will be notified when the update has been downloaded and is ready to install." That message stays there between 10-30 minutes (every attempt to download the software had different timing). Finally, a message pops up stating, "Software Update Failed". (See screenshots attached below in chronological order.)

Before suggesting solutions, know that I've eliminated the following as possible reasons for the issue: 1) I've let my phone die / restart before attempting installations. 2) I've plugged the phone into power and been connected to wifi during every installation attempt. 3) I have 10 GB available on my iPhone, so there's definitely enough space for the download and installation to take place. 4) I've plugged my iPhone into my computer (iTunes) and attempted to update the software via the computer. Spoiler: It still didn't work.

Fingers crossed that I can resolve this without having to go down to the Apple Store Genius Bar for help!