What size sub-panel and wire do I need for my three-season room?
You don't need that big of a feeder, but if you're going to put in a subpanel, you might as well take the chance to free up space in your main panel

Doing an Article 220 calculation for your feeder alone, with 990VA for the lighting, a 1500VA extra allowance for receptacle loads, and 2875VA for a 10A@230V mini-split (12kBTU/20SEER) at 125% (as the largest motor on the feeder), gives us a 22A feeder, far short of even the 60A that 6/3 NM is limited to.  However, I would run either a 1/0-1/0-1/0-2 Al SER cable (good for a 125A feeder) or a 1.5" EMT (with stud shoes, but also good for 125A tops with 1/0 Al XHHW-2s in it) between the main panel and the subpanel.

This allows us to put in a 125A, 24 or 30 space, main lug panel for the subpanel (a HOM2448L125PGC provides a nice, easy package solution if you wish to stick with Homeline), with a HOM2125 for the feeder breaker. Even with two 15A AFCIs (HOM115CAFI in Homeline) for receptacles, and a 15A double pole breaker (HOM215) for the air conditioner, this still leaves ample room for future expansion, which is a good thing considering that your main panel is basically full: you can get 2 spare spots by pulling out the other 40A breaker from the tankless experiment, but that's it, and you'll need those spaces to retrofit AFCI protection as there are no double-stuff AFCIs.

The HOM2125 replaces one of the unused 40A breakers that originally went to the tankless heater, by the way.  Also note that you will need to torque all lugs to specification with an inch-pound torque wrench or torque screwdriver; this requirement was enshrined in Code in 2017 with 110.14(D), in addition to being a good idea lest your electrical system pull a Greg Biffle on you.