Look up a YouTube video on fixing large holes in walls. Get a piece of carpet to hang over repair for insulation purposes

you own or rent?

You really need to post some pics

How big is the hole? What are the walls made out of? Cladding? Brick?

Can we get a picture? It's clearly an exposed wall, but how much insulation is in that wall or specifically between that stud space? Do you have electrical running in that specific section? Own or rent? Do you need to be able to undo most of your work? If you don't care about insulation, you could technically drywall over it; but you're still going to get a very cold wall.

If you aren't able to break the wall open more (insulation batting, hard foam board), then you're really left with trying to insulate what you can get at, and close the wall up.

Fast, sturdy solution: Bolt some plywood over the outside. Caulk the edges. Couple coats of exterior paint. Stuff insulation into the wall gap space. Drywall the inside surface.

Temporary fix: Sturdy sheet plastic on the outside, stapled down. Insulation in the gap. Plastic sheet on the inside.

You can start by grabbing some expanded owens-corning pink foam boards. They come in 2" thick and are easily cut. Also get a can of the water-based spray foam, so that you can clean it up easily in the future if you need to. Cut 2 pieces from the foam board to fit the hole and so you end up with 4" of insulation. Fill any gaps around the edges with the spray foam, and enjoy the lack of cold air making it into your living space.

Temporary solution while you figure out a permanent one is duct tape and a trash bag.

But the most important question has already been asked. Do you own or rent?

Replace it with a window

I Own so any fixes are on me.

This is the answer.

2 inch is overkill though, two layers of half inch or so one at the outside and one neat the inside of the opening will be THE most insulated part of your wall.

The stuff is incredibly light, incredibly easy to slice with a boxcutter, and incredibly incredibly insulative.

Cut it intentionally a bit bigger, put it in diagonally and push it towards being flat.

It will wedge and seal like the dickens as you shove the leaning sides.

Then do another one or so their is air space between them.

Measure your hole before you go and cut with 2 inches spare there.

(These things are cheap, but come in huge sheets that are a pain to transport)

Then stick the spare stuff in the back of your closet and haul it out whenever you want to be constructive over the years.

This is the right answer.

Tip: Bring (or buy) a handsaw (doesn't need to be an expensive one) so you can rough cut the board and fit in your vehicle. Then you can cut it to fit at home.

Installing a window is probably best.

However, to patch it is going to need some work. I do not see any outside photos. My biggest concern is the opening to the elements. Install stucco or sidding on the outside. Install insulation in the wall. Foam board or fiberglass insulation. Sheetrock, tape, texture, prime, and paint the inside.

Edit: Make sure to take out the wood molding in the pic. When installing the sheetrock, make sure it is flush with the rest of the wall so it will look smooth when finished as if it was never patched.

Take measurements before you buy it, then cut it to size in the parking lot outside the store where you bought it. ... I have been known to take a T-square and a handsaw with me to Lowe's to cut timber or boards even though I own a pickup. Sometimes it is just that much easier to carrying material already cut into smaller pieces.

A window would be relatively difficult, time-consuming, and expensive (it will need a custom size), not to mention that it will almost certainly look weird due to size and/ or location with respect to the floor and/or other windows.