Well, your gas bill will surely be less than your electrical bill for the same heating.

But you also didn't give any indication of where you live are you heating this room in Florida or Minnesota? Big difference.

A previous owner added a three season sun room at best. If you are using it as a four season room above the Mason Dixon line, you have to heat it, and compensate for lack of even a crawl space. You don't mention the flooring, but I'd start with heat mat under tile, and use the fireplace when you're sitting out there. It is what it is. A cheap foundation is followed by less "cheap" comfort options.

location?...have you asked HVAC dealer if ductwork can be connected to addition...if not, then look at a small mini split HP system....don't be surprised, they are pricey and may not be aesthetically pleasing to the eye...if you have nat gas, a non vented gas insert for fireplace would work...and certainly would be a less expensive option to a mini split....I will assume from your post that AC is not a concern...yes or no...

We installed the Mitsubishi wall mount in two bedrooms, and they're fantastic. Like cold air returns, light switches, etc, you get used to them.

What about electric baseboard heating? I have a room that was added to the back of my house - the room has 3 exposed sides that are mostly windows. Even though the room is connected to the house hot air heating system, the room was really cold in the winter (I live in NH). So we installed some electric baseboard heat with its own thermostat and it works great. The new electric baseboards are much more low profile than the old style so they're not overly obtrusive looking.

all electric cheap to install but expensive to operate...

IMO

Consider a vent-free gas log set. For about $500 you could heat that room easily and cheaply (they are virtually 100% efficient). We have an open plan great room, dining area, kitchen and foyer (well over 1000 sq ft) and a small vent-free gas log set heats it up nicely in no time. The house is located in western virginia and we use it when the temperatures get down around freezing (we use an air to air heat pump above that). Its wise to have your fireplace damper very slightly open for some air exchange as these fires do produce some carbon dioxide (NOT carbon monoxide - they have sensors built-in to cut off the gas supply if they detect carbon monoxide). They also produce some water vapor so you have a built-in humidifier!

The mini split is the best option in my opinion. It would provide both heating and cooling. The heating cost would be less than baseboard electric heat. But mini split installation costs tend to be expensive. I suggest you get some quotes and consider it as an option.

While exploring some options for heating for a possible future addition, we came across this product: https://climateright.com/

We haven't used it personally so I can't give a true recommendation. However, it might be an option to explore.

If you need heating and A/C in this room, my suggestion is to consider a wall mounted PTAC heat pump. These units are what you often find in motel rooms. Easy to find models are of two types - the heat is produced via resistance heat (like an electric baseboard, but more expensive to operate) or via heat pump operation (less electricity consumption for heating).

If just heating, I would get a wall mounted vented gas heater.

The disarray in that room suggests it's for "throw it in the junk room" purposes. If so, maybe it doesn't need heat? If not, it needs some help. Good luck.

Our Heat n Glo gas insert has a thermostat.

All of these heating/cooling products are options. The heat pump is a great option. With tall ceilings, a large ceiling fan that can be used to "push" the heated air BACK DOWN into the space is WELL worth the effort. It can then be used as a cooling fan in the summer months.

There are two elephants in this room that no one is talking about: 1. INSULATION. 2. Chimney. Insulation: How much and what are you working with? How old is the addition? By the colour/appearance of the stone, I'm guessing 90's.

My first thought is to rip out/close up the chimney. It is a big hole in your house that is leaking heat. Rip it out and you will feel the difference. Like hole in your roof or a hole in the wall, a chimney is one of the biggest sources of energy loss a home can have.

When you remove the chimney, have a look at the insulation in the walls. Take a look to see how you can improve the heat retention simply by upgrading your R value in the walls. Something from the 90's (20+ years old now) may be considered obsolete by today's standards.

Once you have corrected BOTH items, now you look at heating/cooling options. It is a waste of money to add MORE heat/AC in a space that is performing poorly in the first place. Shore up the deficiencies in the system and then see what needs doing once you have increased the efficiency by plugging the holes.

There's no use adding more water to a bathtum if it has a hole in the bottom.

You could get an electric insert for the fireplace with a nice fire display with or without heat. There are other options as well such as entertainment centers and stand alone units. If you are going to use it for heat, get one with infrared heating for sure. Good idea to have its own electric line and breaker if using it for heat. Display only doesn't use much electricity.

Electric infrared is not expensive to operate, they do a really good job of heating, probably twice as good at heating than coil style heaters. A 1500w infrared would heat 500 sq ft easily.

I had the same problem. I bought an electric fireplace, has a thermostat, can heat up to 1000 sq. ft. Works great.

"Electric infrared is not expensive to operate"

Except where electricity is VERY cheap - as it is in a limited few areas of the US because of hydroelectric or cheap coal electricity - this is usually not true and often it's the most expensive way to heat.

If you didn't have a toddler, I would suggest ripping out the fake fireplace and putting in a woodburning stove (assuming your municipality will allow it). In Michigan you must have millions of trees to harvest---it's great exercise to find the trees, saw up, get them home, stack to season, saw into 16 inch sections, split, stack, etc.

You can often find free wood after a big storm too.

But if you cannot do a woodburning stove, the Mistubishi mini-splits is the way to go. Sad that you are on a slab--in-floor heat is wonderful. Also look into the insulation in walls & ceiling!

if you have nat gas, I still believe a gas log insert would be a possibility....the good ones have a remote control with thermostat....at least take a look at a dealer...not a big box store either...

IMO

The gas fireplace we have can use natural gas or propane, a pellet stove can come handy too in the cold weather.

We closed off a wood burning fireplace, insulated a bit in the chimney, and put a little electric fireplace in the fireplace spot it heats our now finished basement space just fine. Cost less than $150.00.