I also agree with roy smith's method.

1) There is a relation between Milnor number and the total number of vanishing cycles, you can find this in Deligne's paper: La formula de Milnor, SGA 7,Expose XVI. In Proposition 2.1, he gaves a identity between Euler-Poincare characteristic and summation of milnor numbers.

You can find a relationship between milnor number and degree of local Chern class in the following paper(and its' ref).
2)F.Orgogozo's paper: Conjecture de Bloch et nombres de Milnor.

I hope this two papers can help you in some sense.