On the numerical value of $i^i$ and Historical notes on the relation $e^{-(\pi/2)}=i^i$ describe how these accurate computations can be performed with logarithmic tables.
Euler described how he arrived at the identity in a paper read at the Berlin Academy in 1746, giving more decimals (13) than in the letter to Goldbach. A later calculation by Gauss computed 35 decimal places. Euler did not present his computation, but Gauss did [source].