For generic determinantal varieties, one log resolution is what is classically known as the "space of complete collineations". In the following article, Vainsencher computes the integers you want (although he uses slightly different notation than is common now).
MR0738261 (85f:14053) Reviewed
Vainsencher, Israel(BR-FPN)
Complete collineations and blowing up determinantal ideals.
Math. Ann. 267 (1984), no. 3, 417–432.
14M12 (14N10)
Edit. Also, before I was aware of Vainsencher's article (which I learned of thanks to Michael Thaddeus), I also computed these discrepancies in my preprint on Kodaira dimensions of the Kontsevich moduli spaces of genus $0$ stable maps to Fano hypersurfaces. So that is another source. Surprisingly, generic determinantal varieties turn out to be good models for the singularities that arise on Kontsevich moduli spaces, at least for "low" degree / homology class of the stable map (and one can use this to prove some Kontsevich spaces are terminal / canonical / log terminal / log canonical).