How to connect a wireless monitor to mac mini so that it is the only one?

Question

macOS

Given:
Mac mini mid 2011 with OS X 10.9
Apple TV Gen 3
HDTV with miracast
wi-fi support
Task: I have the above computer. There is also a desire to put it in a closet and leave only the power cord. Mac stubbornly requires that at least one wired monitor (via HDMI) be connected to it, otherwise it disables wireless ones. Deep in my brain I feel that it is real and without wires, but I can not understand how.
What I already tried:
1. standard OS X tools: using airplay, you can mirror images on apple tv, and therefore on TV, but the problem is that when HDMI is disconnected from the computer, the airplay image also falls off. Making the mac think that the monitor is connected to it by leaving an empty string is not an option: after about a minute, the airplay image falls off + permanent artifacts.
2. AirParrot + Airplay: actually the same, but more interesting here. After disconnecting HDMI, the image does not fall off, but the keyboard and mouse fall off. In English-language forums they write that it should be so, because mac disables them as unnecessary, thinking that the monitor is still not connected, if the mouse and keyboard worked - this option would be ideal.
3. Miracast: quite a perversion, I thought, but decided to try. The same. Keyboard + mouse fall off.
I understand that it would be ideal to make the mac think that a wired monitor is connected to it, then all three methods would work. Another option is to force him not to turn off the keyboard and mouse, but I did not find how to do it.
Obi-Wan Kenobi toaster, you're my only hope, please help.

Answer 1

I think only something like this will help you
www.amazon.com/CompuLab-fit-Headless-computer-disp ... enable-awesome-vide...

Answer 2

it turned out to be easier to hang a poppy for a telly and buy a wireless keyboard + mouse.
Thanks to all.