There really is no solution to your problem. The largest issue is HDMI signals are not compressed data. This why the cable lengths are so restricted.
When you do the math a 1080p at 60 frames is over 4Gbps and can go much higher if you use high color depth . There is no wireless technology that can come even close to that.
You are going to have to run cables no matter what method you choose.
There are physical cable converters that allow you to use the same cat6 wire that is used by ethernet but it is not running ethernet. It has fairly limited distance constraints. There is no standard for this so you see many implementation from many vendors. It is cheap but does not work all that well from what most reviews say.
You can run it over ip but almost all the product you find on the market...especially stuff sold to end consumers....is a massive hack. What the do is try to compress the data on the fly so they can put 4gbps signals into a 1gbps data stream. To do this well takes expensive hardware. The units selling for $100 are close to worthless, they will be fine on data that is not changing much but anything like live sports will become a pixelated mess. I have seen some high end commercial stuff that uses 10g networks so it can send it uncompressed. They do not even put a price on their product so you know it can not be cheap.