some insulation in the walls, then peg board ( like 1/4" masonite with holes all in it ) , then a layer or 2 of drywall and some moving blankets all around . will kill alot of the sound. thats how we build my buddys vocal room in his house for recording. ( raised floor of course) room is dead on inside and VERY quite on outside. NOW i know this is in a shop so blankets might not be safest thing around all that heat ? but i dont know how much room you have? but a couple alternating layers of a softer and a more dense materials will kill ALOT of sound. they have a product too i forget the name , its like this flaky sound barrier stuff, kinda looks like a super thick green ceiling tiles. really good at killing sound BUT i dont know how safe it is to work with ... also the way it kills sound is by turning the sound energy into heat energy, not much but still heat... i dont know if it can effect room temps but probably dont want it any hotter in your forge? haha just some ideas to play with hope they help. i love sound and audio, kind of another hobby of mine, way too much to get into on here, sound can get crazy almost pushing boundaries of physics. also a nice lattice or arbour in front will make your shop look a lil nicer add privacy and help with sound. or open doors but hang moving blanket infront of open doors ? but it sounds like you are working within the law and also within common decency , so i hate to say stick it to her... but some people you can never make happy, most of my nieghbours are older and i just find they dont like me, but im younger guy, big dog, tattoos, nice car ( not that nice, i wish i had a german car) , they have an impression of me thats wrong , if i had the german car id be undersame impression hahaha hope some this info helps and it all works out, maybe make her something ? then she will see what all the noise was for ?