I don't believe mowing is going to get rid of the milkweed.

Question:  if you eliminated the milkweed, is there grass underneath that will come back?  Are you better off killing the whole sod down and starting over?  Maybe some glyphosate + 2,4-D to kill the sod down completely, and reseed.

Another alternative that has crossed my mind - spraying with Distinct/ Status.  I haven't checked the label - it might be an option to look at.  That would allow you to keep the grass that is currently underneath, depending on how much is there.