According to the dynamo theory,
[...] the magnetic field is induced and constantly maintained by the convection of liquid iron in the outer core. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the Coriolis effect caused by the rotation of the Earth. The Coriolis force tends to organize fluid motions and electric currents into columns (see also Taylor column) aligned with the rotation axis.
To compute the magnetic field, or the electric current density, one should then be able to solve the extremely complex nonlinear magnetohydrodynamic (MHD) equations for an electrically conducting fluid undergoing thermal convection in a rapidly rotating spherical shell.
This is quite an ambitious goal: it is because of this that you couldn't find any estimate of the electric current flowing in the Earth's core (more exactly, it would be current density).
It must be mentioned, though, that there have been some important numerical studies, one of which was even able to predict the geomagnetic field reversal.