You did not mention in the original post that the other computer had no monitor. Would have been helpful to have that info.
That is because the goal is to have no other computer. Just a keyboard with a long range. It's why I don't mention a second computer in the post at all, and just ask about sending commands over the network.
The overall system is a rackmount computer in a network closet with the video output going to an HDMI matrix. The HDMI matrix then feeds the output to one of several TVs around the building. I would like the ability to sit at one of these TVs and use it like a normal computer. IE typing on a normal keyboard and normal mouse with the TV acting as a display. The problem is that these displays are too far away from the computer case for a normal wireless keyboard/mouse to connect.
As I was familiar with apps that allow for mobile devices to act as mice over ethernet, that was the first place I jumped to as if there was a keyboard that connected via 802.11 then it's range would be the range of the wireless network instead of the 10ft or so of a normal keyboard.
A USB over ethernet bridge with a bluetooth dongle presents a decent option. That is what I am looking at now, but it still seems like there should be a better option.
Again to reiterate: I don't want a second computer. If someone was to sit down at one of these screens I want them to be able use it like you use a normal keyboard and mouse.