I finally came across Fire TV after searching and comparing different applications to stream media content from iOS devices to this inconspicuous and must say that I am very happy. The price is absolutely justified. I managed to get on the TV screen with this one both audio and video (mirroring).
An important note about this: need to activate the mirroring one on iOS devices in the control center (setup menu in which you run through from bottom to top passes) and there change the slider. Otherwise, if necessary, only audio is transmitted. The out cost me some time and nerves.