That's a pretty narrow interpretation of "does not have a processor". Keyboards and mice have processors. HDMI adapters have processors. Wireless transmitters and receivers have processors. Linking all those processors together requires - you guessed it, a processor. By the time you get there, the cost difference between a dedicated micro processor chip like a mouse or keyboard controller compared to a fairly powerful SOC like the ARM chip on a RaspberryPi is down in the "cup of coffee or two" range.

You could pretty much do exactly what you describe with a RaspberryPi and some software - instead of two or three dedicated mouse/keyboard/monitor tx/rx pairs, you plug those peripherals into the 'Pi and use WiFi and software. (Note: a RaspberyPi isn't the ideal candidate for this, they made design decisions way back that limit the network throughput, so I don't _think_ you'll get full HD video over the WiFi adaptor because of the way it's sitting on the relatively slow and congested USB bus, but that's not a fundamental flaw in the idea, just an implementation detail the 'Pi designers chose to make back in '12 or '13 that you wouldn't need to follow now...)

For very little more than a Raspberry Pi (plus power supply and USB WiFi adapter), you can get many "mini pc"s now - I've got an Ainol Mini PC which is an entire Windows 8 machine, with WiFi/USB/HDMI/power-supply(including an internal battery good for ~10hours) - and you can get them for around $80US ex China now. I'm not sure how much more "compact/cheap" you're thinking of, but I wonder how big the market of people who would use your hypothetical gadget but only buy one at a pricepoint somewhere under $80 is - when the alternative is a full blown, if somewhat under powered PC.