HHhhmm, appears that nobody has warned you about stainless steel being a " bit on the grab it " type of material when it's being tapped.
If you want constant and continual success at tapping into stainless steel with a 5/16th x 18 tpi tap, or any other tap that you want to use, then just make sure that you drill the hole ONE METRIC sized drill UP, than what the charts recommend.
For the 5/16th x 18 it calls for a 6.6mm drill, GO FOR the 6.7mm drill, OK you might loose a bit of the thread form, but it more than makes up for NOT having to attempt to try and remove a broken tap.
OK, the drill charts call for certain sized drills, which will give you a 75% thread form, so a drill size UP, might equal a 72% thread form, big deal, still no broken tap, is definitely worth that slight loss of thread form.
By the way, any tap will now work, gun, spiral, bottom, even an old carbon tap will still be OK, if you drill the hole one size up.
regards greenie.