Dimensional analysis of integrated white noise process
The problem comes from the dimension of $\sigma^2_i$, which you assume to be $[I^2]$. That's, however, not the case if you define

$$E\{i(t_1)i(t_2)\}=\sigma^2_i\delta(t_2-t_1)\tag{1}$$

Note that the dimension of a Dirac delta impulse is the inverse of the  dimension of its argument, which is $[1/T]$ or $[Hz]$. So you get for the dimension of $\sigma^2_i$ $[I^2T]$ or $[I^2/Hz]$. You can also see this by looking at the power spectral density (the Fourier transform of $(1)$), which is simply $\sigma^2_i$, and the dimension of which must be $[I^2/Hz]$. With this dimension of $\sigma^2_i$, the dimension of your result is correct.