Gaming carts are furniture objects that have found their place in hospitals, typically in Pediatric Wards. They can be wheeled out of a storage closet and into a patient's room, and easily gotten out of the way when not needed. This model here, the FLE Arcade Cart II by Fully Loaded Electronics, takes its design cues from other types of medical carts; it has the narrowest-possible footprint and is touted for being easy to clean.
This Hygia VHA Gaming Cart by Parity Medical has a foot-pedal-operated height adjustment system and a tiltable/extendable screen mount, so it can be used at either chair height or bed height. All of the cables are enclosed, making it easier to clean than the FLE above.
Another design offered by Fully Loaded is this Go Kart, which securely encloses the gaming console, presumably for situations where it might otherwise go missing. It's sad to think that someone might steal a gaming console from the Pediatric Ward of a hospital, but I suppose that's the world we live in.
None of these units list prices on their websites; you must instead contact them for quotes. But I found an article from a local news network in Kentucky that told of a charity organization enabling the purchase of a Go Kart unit for the Pediatric Unit at the Mercy Health - Lourdes Hospital in Paducah. The precise price of the unit was not listed, but the article mentions that "The purchase was made possibly by a $3,500 grant from the Honorable Order Kentucky Colonels."
Furthermore, the Charity Gaming website, which solicits donations for gaming carts, has $5,000 set as the default donation.
Gaming carts for hospitals doesn't appear to be a terribly competitive space; the manufacturers above were the only three I could turn up. I think there's some room for innovation and price-competitiveness here, or at least a senior thesis project with an unassailable goal: Making these more affordable.