Many maid cafes allow you to play board or video games against one of the maids for about 500 yen for 3-5 minutes. Few let you be the maid.
Well, technically, it's not a cafe, and you're not really a maid, but why pick nits? The Melty Cure website has a link for a handful of Flash games that are kind of fun to play. The first game is "Walking Reversi", where you guide a maid to turn all the pieces red. The second is a trace-a-route game to have the maid reveal a picture (mess up the route and you have to start over). Third is a reaction time game where you click to pound a batch of mochi while trying to avoid hitting your partner on the head. Fourth is another reaction time game where a cat tries to catch a waving cat toy. Then we have LT-RT, a symbol matching game similar to Tetris (4 or more of each symbol in a row makes them disappear). Panpan Waroon has you clicking on balloons in numerical order to see if you can beat your best time. And finally there's the gramophone (click on the record to play a song.)