I assume your gap is on the indoors side?
Regardless, if air is filtering through into the living space, there must be a gap outdoors ( external chimney) as well as indoors.
I recommend you check out the caulking along the chimney to the exterior siding as well as addressing the interior gap. Since this gap inside is directly adjacent to wood trim, I also have to assume that excessive heat is not a factor. Is the metal you are referring to an insert trim of some type? A pic would be a big help here.
If the assumption are right and the gap is 3/4 inch wide, I would use a caulking backer in the gap before using a good grade painter's or acrylic caulk as a topper. The caulking backer is a round foam solid tube that can be cut to length, inserted into the gap to seal it and hold the caulk. After the caulk has cured, it can be painted to match the trim. If for some reason you are concerned about high heat in this area, there is a fire rated caulk made especially for gaps in fire or common walls. Readily available at any box or hardware store from a number of manufactures.
Ok after reading your comments, and learning that the black vertical sides are slate, I would still do similar to above. Perhaps after injecting larger amounts of fire rated caulk as deep into void as possible, then use the solid, tube style caulking backer to fill the majority of the larger, deeper gaps, followed by using some paint-able silicon/acrylic mix caulk.
Before you apply the finish caulk, run a nice straight line of painter's tape up the front of the slate even with the edge of the wood trim. When you fill over the foam backer with the caulk, use a small putty knife to draw the caulk smooth and flat. Immediately remove the tape while caulk is fresh. This should result in a nice crisp straight line. Let the caulk cure for 24 hours. If there is some shrinking or cracks, no problem, simply repeat the process with new tape and another coat of caulk.
Alternately, you could also use a setting type drywall mud, such as durabond 30 or 45 to fill the deeper gaps etc. Then again use a second coat to create a smooth straight extension of the wood trim to the slate.
Either method will result in a neat, easy to paint fill that should stop your air infiltration problem and make the finish look nice.
The nature of task will depend on the exact type of wall. It would be best to wait for the inspection report before deciding anything.
If you have a cavity (for example) then it might be best to hire a professional. If you don't have a cavity then you need to add some insulation to the inside walls. You can get plasterboard with integral insulation which you can attach directly to the existing wall, but if you need more insulation you'll have to use thicker battens to allow for this. This will reduce the internal dimensions of your rooms. This is a job you can do yourself - but it is a big job.
Attic insulation is something you can do yourself too. The inspection should recommend what level of insulation you need to install.
Best Answer
Problem #1: When installing insulation, you should think “envelop”. That is to say, just applying slabs of insulation to exterior walls does not “help” the energy conservation of the building, if there are gaps between the slabs, around the edges, etc. there will be NO insulation qualities.
Problem #2: Moisture Barrier. I’m sure there is an existing moisture barrier. (Note: moisture barrier is different than vapor barrier. Let’s discuss moisture barrier first.) Moisture barriers are installed to keep OUTSIDE moisture from penetrating the exterior walls and causing dryrot, mold, etc. When you add another exterior layer, like insulation, it requires another moisture barrier. If moisture gets past this new moisture barrier it can get trapped and create dryrot, mold, etc.
Problem #3: Vapor barrier and Dew Point. Vapor barriers keep vapor from exiting the living spaces and getting into the insulation. Once in the insulation it can change into moisture through a process called “Reaching it’s Dew Point.” The placement of vapor barriers is important because vapor can change into moisture at the Dew Point and create dryrot, mold, etc. in the walls and ceilings. When there are two moisture barriers, this will often occur between them, depending on the amount of existing insulation in the wall.
Dryvit is a manufacturing company that has a system of installing insulation on the exterior of walls and then troweling on a coat of waterproof plaster. If you google it, you’ll see about a hundred lawsuits for these identical problems.
Be careful adding new “systems “ over existing systems.