The recreation facility needs a weight room/gym. Maybe with treadmills as well, and a couple of professional coaches to keep things safe.
Oh! Where's the hot spring?
He could actually do well to make his Duchy into a recreational resort with different towns having different trim levels. Top-level for heads of state, being intended to give them a place to get to know each other away from the pressures of politics. There could be a Camp David type of area for high-level meetings that would be held secretly, with the presence of the different leaders announced publicly as simple recreational activity. A fully staffed hospital would be maintained on-site, and the city would be a no-fly zone. The different countries and territories would agree to certain conduct, while also paying a regular fee for access. The fee would help cover the expense of such facilities.
Other recreational towns would host similar facilities, but with a lower level of attention to customer service. There might be more facilities for children, including sporting facilities and some theme park type of environment. It would be open to the public, but the price of entry would be high. Nearby towns might be more dedicated to retail sales for anyone who can afford the goods, as well as classy restaurants. Other offerings might include concerts, theater, and opera.
A lower trim level might be offered in other towns, with ordinary-level lodging, shopping, and dining. People could buy a vacation package. Familiar sports and games, dances, minstrels, and such would be offered, with facilities that require much less maintenance. There could be small game hunting and fishing in stocked lakes and game reserves. The fish, animals, and birds provided would be raised in a farm setting for release into the hunting areas, keeping populations high enough to make it easy to find prey. It would be like a set of towns with permanent festivals going on.
All of this would require a thriving middle class in the surrounding countries to provide a steady customer base.