I had to look this up because I was confused about how difficult this would be earlier.
Good news is that if we can figure out how to make Gomril wire, from there on we've pretty much solved all the problems, from there on its just finding ways to apply our solutions in different contexts.
There are a couple of steps to do it.
First you make the metal wire. This has since prehistoric times been done by essentially squeezing a wire through a small hole called a die. This is going to be extremely difficult as the force needed is proportional to the change in radius and the strength of the metal that is being drawn into a wire. Given the ridiculous strength of gomril this will either require an unbelievable amount of force or an incredible number of different repetitions through increasingly small dies. Another problem that hardness adds here is that because hardness is the resistance to deformation. Steel wires typically require certain alloys in order to be drawn, since gomril is harder than steel we either need to figure out an alloy that could work or else we alter the ductility in another way such as working it at extreme temperatures, since for earlier reasons the drawing is likely to be a extended problem this has its own problems.
One reason this may be a stumbling block for other dwarves is that just as a huge amount of force is exerted on the wire, this means a similar amount is exerted on the die. Because of this you cannot effectively draw wire from a die made of a metal weaker than the wire as it will rapidly deform and even when they're the same strength the dies will have a short lifetime. Because of these, having Adamant in order to make the die may be a necessary step inorder to make Gomril chainmail.
I think we're probably going to end up creating a Gronti just to draw it, seems to be the only way I can think of to get sufficient force in this pre industrial society.
Next step is forming the wire into loops. This is usually done by bending the wire into a coil around a cylinder of the diameter. Ductility is a problem here again but if we've reached this step without doing something like casting the wire then this is probably a solved problem. Cutting the coil into loops to get them off the cylinder may also be a problem hypothetically.
Last step is actually weaving the loops into a weave and probably welding the open ends of the loops. At this point the problems we'd be facing are mostly the same as before, fine ductile manipulation of thin loops of gomril without them shattering. Applied heat to melt the ends of the loops without melting other loops ( I feel like our best option is probably not doing that and applying some kind of gomril solder). This is absolutely the most fiddly bit and depending on how many times we had to repeat the drawing process the most time consuming however Snorri should be familiar with it.
Wikihow actually has an article on this for larpers and people interested in home crafts.
Whether you're a LARPer, a cosplayer, or you're heading out into battle, making chainmail is a useful skill to have. If you've never made chainmail before, there are a few basic weaves you can start out with to make shirts, jewelry, or...
www.wikihow.com
Although they recommend that you just buy your wire rather than draw it yourself.
E:
Or odd places.
Why keep lengthening the research backlog it just adds more things that people complain we're not doing.