How do I get Apple TV 3rd Gen to connect to new Wi-Fi settings with no remote and a TV too far away for ethernet cable connection?
Apple TV 3rd generation remotes actually use Infrared (IR) technology, so you should be able to just use the Apple Remote. The only reasons your remote may not work with the Apple TV are:


A dead battery 
you have previously linked the remote to another device (e.g. a Mac)


So, if your Apple Remote is not working with your Apple TV you can replace the battery and/or link it to your Apple TV.


To replace the battery refer to How to replace the battery in your Apple Remote. Once you've replaced the battery, if it doesn't work with your Apple TV, then try the next step.
To link the remote to your Apple TV you can just press and hold the Menu and Right buttons simultaneously for five seconds.


As an aside, you may also find that using the same SSID name and password for your new Wi-Fi network (as you did for your old one) means that your Apple TV will be able to connect to it.