Seems like the broken shower arm is a fairly common problem. WTF? no 5/8" nipple extractors on the market?
I saw this suggestion from another post and it worked like a charm, after half the day was spent trying to use 1/2 inch nipple extractors (Harbor Freight/Home Depot), hacksawing the broken threads out, nothing was working.
Buy an Irwin or Hanson EX-7 Spiral screw extractor #7 Part number 53407.
I found mine at Ace hardware in WA state. Call ahead as its not a very common part, note that Home Depot did not sell these. If the pipe on the other side of the broken threads is long enough, just tap in the extractor and twist with a crecent wrench, this tool will pull the broken threads right out. My job was such that there was a 90 degree turn right behind the broken threads, i had to cut the #7 extractor with a dremel, not as bad as you think, it only burned through 2 heavy duty cut off wheels, took about 5 minutes to get thru.
Do yourself a favor, track down this tool and save yourself hours of frustration.