You need professional advise. Someone on site to poke around and see what your options are for insulation, how tight the door, what's under the floor. I've had a similar 'room' is a couple of homes and the construction is just not as solid or buttoned up as the house. You would need to fix that in order to not be heating the outside.

Its called a breezeway for a reason they are really just an unheated walkway. I think if you have agarge that would probably be easier to heat and get some good grow lights for that area.. From what I see in the pics this is going to be an expensive redo almost like an addition

Hi, Emily,

It's an awkward space to be sure, but I wouldn't throw in the towel just yet. Right now, it appears the space functions as a connection from your home to the outdoors and a poorly performing mud room/catch-all storage area. I don't ever see this area as an inviting area to sit and relax so I'd ditch the furniture. I'd suggest you consider sub-dividing the space into two spaces--a proper mud room with storage for coats, boots, umbrellas, dog gear, etc. and a bench where you can sit down and put shoes and boots on/off. The other space could be a potting shed/greenhouse (albeit without much natural light) optimized for plants.