I think I see now what is happening.
Mathematica does not recursively define the D operator using the chain rule, presumably because this is too slow. It does some pattern matching on subexpressions to see if they contain the variable of differentiation, and subexpressions which don't are treated as constants; so my pattern for D is never applied.
The way around this turns out to be to tell Mathematica explicitly that
Subscript[w, i]
is not a constant.
My pattern now looks like this
Unprotect[D];
D[Subscript[x_, i_], Subscript[x_, j_],
NonConstants -> {___, Subscript[x_, i_], ___} ] := KroneckerDelta[i, j];
Protect[D]
And I have to apply it this way:
D[k * Subscript[w, i], Subscript[w, j], NonConstants -> Subscript[w, i]]