Yes indeed, n corresponds to king, as we need two kings of a different color to have a legal chess position.
Also, as you give, h = pawn, as otherwise we would have a pawn on the first or last rank, which would be illegal.
It cannot be a queen, as both kings would be in check.
It cannot be a knight, as the "i" on a2 would give check. There is no legal way where "knight a2 check" is the last move.
It cannot be a bishop, as the "I" on a5 would give check. There is no legal way where "bishop a5 check" is the last move.
Therefore, i = rook.
We see that the king on c3 is in check by the rook on c1. Therefore, g cannot be a queen or a bishop, because there is no legal way to obtain the resulting position.
So, g = knight.
The "K" on b8 should not give check to the king on b6.
Therefore, k = bishop and t = queen.
[However, the resulting position seems to be impossible to achieve too...
As far as I can see, the puzzle has no solution.]
Edit: the resulting position is indeed possible, see the answer of Glorfindel.