Volume current density, $\mathbf{j} = \rho \mathbf{v}$, can be interpreted as the elementary charge flux across a surface. This interpretation comes from the integral balance equation for electric charge, that reads
$$0 = \dfrac{d}{dt}\int_{V} \rho + \oint_{\partial V} \rho\mathbf{v} \cdot \hat{\mathbf{n}} = \dfrac{d}{dt}\int_{V} \rho + \oint_{\partial V} \mathbf{j} \cdot \hat{\mathbf{n}}$$
for a volume $V$ fixed in space, stating electric charge conservation (you can't destroy or create out of nothing, as you can't destroy or create mass in classical mechanics).
Thus, dimensional analysis gives
$$[\rho] = \frac{[\text{charge}]}{[\text{length}]^3}$$ $$[\mathbf{j}] = [\rho] [\mathbf{v}] =\frac{[\text{charge}]}{[\text{length}]^3} \frac{[\text{length}]}{[\text{time}]} = \frac{[\text{el. current}]}{[\text{length}]^2}$$
while surface and linear density comes from one or two integration in space, and thus
$$[\mathbf{k}] = \frac{[\text{el. current}]}{[\text{length}]} \qquad , \qquad [\boldsymbol{\lambda}] = [\text{el. current}]$$