Did you use a protein spotter on it? Watch your PH a lot of them can be quite high. Not good for natural fibers, especially dedicates like silk.
Test for color difference or lightened areas. Its no good to remove the milk but leave a halo when you used the product.
If it were stiff from the cleaning then all of the rug would be stiff (very possible with silk) but if its more so where the milk was spilled, then I don't think the residue is completely gone.
Being silk, I'd call the client and see if she is willing to take on some risk of you spotting the milk. No point on you taking on the liability unless you spilt it.
