While researching this problem, I came across a cheap device called the Anycast M2 Plus.
http://www.amazon.com/gp/product/B06XYJPVR1
I dismissed it since it was under $20 and assumed it was just a ripoff joke that wouldn't work at all. After not being able to find ANYTHING else that could do what I wanted I figured, what the hell... I'm only out $20. So I ordered one.
It arrived a couple of days ago and I immediately unboxed it and plugged it into my television. I was extremely surprised to find that not only could I easily connect to it with my Android phone (LG V10), but also my work phone (iPhone 6). Screen mirror working fine on both devices. Not only was I able to stream video from my phones to the television, but I could stream from my wireless connection to the phone to the television. I'll try to connect to it from my laptop tonight just as a final test.
According to the documentation, you can connect the Anycast directly to your wireless network, but I did not as the location where I intend to use it doesn't have wireless available. You basically connect to the Anycast via WiFi on your device. Once you have established a connection, you use the built-in connection software on the Android or the iPhone, type in the password displayed on the television and you're in business. So far that seems to be all there is to it.
Obviously I haven't had much time with it yet, but I've been very pleasantly surprised so far. Especially for under $20. I'll post an update once I've had a chance to connect my laptop to it and try it with other people/devices.