OK, so try resetting the device (you indicated that you did a Reset All Settings, but I'm thinking you haven't done a device reset): Hold down the Home and Power buttons at the same time and continue to hold them down until the Apple appears (up to 30 seconds).
Once the Lock screen displays, try the "Hey Siri" command (you were able to get a checkmark on all of the Hey Siri commands when you set it up, correct?).
If that works, then you are all set. If not, then you may have to Erase All Content & Settings and restore from an iCoud or iTunes backup. Just an FYI, I erase and restore my iPhone 5s fairly often for testing purposes. I usually use an iCloud backup to restore, and the backup doesn't take long, but it does take a little time for the media to re-download. I plug it in when I'm doing a restore, and do something else for an hour or so. It is usually completely done by then. You can use it while it is re-downloading your media. It's just a good idea to keep it plugged in to conserve battery.
If you decide to do the Erase and Restore, I would suggest that you set it up as a new phone the first time around (not from the backup) just to see if Hey Siri works after being set up again. If that works, then Erase it again and then restore it to the backup.
If Hey Siri doesn't work after setting the device up as a new device, then you would need to take it into Apple to have it looked at.
If it does work after setting it up as new, and then stops working after restoring it to the backup, then your data on your device is corrupted in some way, and you would have to set up the device as new again, and then manually download your media (or sync it), and manually set up your device again from scratch.
Cheers,
GB