Can't tell you anything about the counterstamp but it is Large Date, Small Stars, small letters with the hub style of 36 with small letters in the outer inscription and large letters for the denomination. N-4 R-2. There are seven varieties in 1834 and only two of them are small stars (N-3 and 4). The small stars have the sixth star pointing just left of the point of the coronet. On all of the large star varieties it points well to the right of the tip of the coronet. To tell 3 and 4 apart look at the point of leaf below the E in AMERICA. On 3 it is right of center of the E, on 4 it is below the right edge of the upright. I never pay too much attention to the large and small letter reverse distinction because the letters are pretty much identical in size. The only real difference is in STATES the TAT is more widely separated at the bases on the small letter varieties. It can be difficult to tell if you don't have known specimens for a side by side comparison. And since the small letter reverses are also only paired with the small star obverse it is easier to use the stars to identify them.
I don't see this one listed in Brunk. Also, Kovel doesn't list this mark for a silversmith. That's not unusual though. Given the size and style of this mark, I suspect that it was perhaps someone who worked in the jewelry trade. The "I" often represented a name that began with the letter "J". The crude letter style makes me think it's an 1830's and likely an even earlier-made stamp. In other words, the stamp could have been applied to this host coin, many years it was originally made. Doing a cursory search, I found two possible makers with JWH initials, working in NY City between 1827-39. More clues will be needed to ID this puppy ... Good luck!
BTW, there are likely many other possible issuers in other cities. I simply found a few prospects in NY City. The crudeness of the letters suggests to me that the stamp was made in the teens or twenties. Of course, it could have been the work of a novice/apprentice, too. Just an afterthought on my part. I'll be curious to see what Bruce says ...
