It’s going to be complicated, but doable. First, I’d look for obvious problems: 1) back-to-back electrical outlets, 2) party wall stopping at ceiling and sound traveling over the top of the wall, 3) wall stopping at corner and sound traveling around the ends of the wall, and 4) wall sits on floor and sound travels under floor (concrete slab or wood frame).
1) Look for electrical outlets (phone boxes, switch boxes, etc) on your side and then measure other side to see if they matchup. If so, you’ll need to separate them and install insulation to isolate the boxes. If this works fine, but if not and it takes a larger retrofit, then make sure these things are separated during the remodel.
2) If the wall stops at the ceiling, (you can lift a ceiling tile and peek in...use a flashlight) you’ll need to extend the wall up to the roof/floor above and cover with sound isolation board. This is usually a very big problem with suspended acoustical tile ceilings. If this is the case, a simple (and somewhat limited success) is to add batt insulation over the top of the wall and then completely insulate both ceilings. (Thicker the better) Probably the insulation has been moved over the years when they installed new phones, data lines, etc.
3) Often the wall ends with no insulation in the corners. This allows sound to travel around the corner. A simple solution is to install 2” (or thicker is better) rigid insulation (you can pretty-it up by covering it up with fabric) and installing a 12” wide strip in the corner going both ways. (Thicker the better.)
4) Wall framing is usually just installed dry (no caulking or sealing material) to the floor. Remove the base and caulk both sides of the wall where it meets the floor. Also, if it’s a wood frame floor, sound could be traveling in the crawl space. This is more complicated, but a wall needs to be built with sound control.
You have two issues: A) airborne sounds, (which is solved with sound control material, (which has a STC rating,) and B) solid borne sounds sounds, (which is solved by separating the wall material in one room from the adjacent room.)
A) Airborne sound can be stopped using STC rated materials. Basically, it’s a combination of sound absorption material and mass. (You can google this and find lots of ideas.)
B) Solid borne sound is more difficult, but basically you are isolating one side of the wall from the other. Usually this is done by staggering studs, building two separate walls, etc. What you have now is basically a “drum.” Your wall has two “skins” on it and sound travels through it like a drum.
I like @Harper ‘s idea: get diagnosed as a disability and federal law says they have to spend up to 25% of their budget (that’s company budget, not construction budget) before its declared “too expensive “. If they terminate you for that, I’d love to be your agent. We’ll both retire to the Bahamas.