The "razor that is hired" refers to the invaders. The implication is that they are mercenaries, but a mercenary is one who does the work of another. In this case, Assyria is doing the work of the Lord, and they are, in essence, his mercenaries, and therefore aptly his "razor that is hired." The shaving of the hair is for humiliation. Both Gileadi and the NIV render the "hair of the feet" as the "hair of your legs" probably attempting to retain some sense while avoiding explicitly discussing the genitalia, for which "feet" was a euphemism.
"The humiliation and slavery that will befall the people is represented in verse 20 by the razor cutting off their hair. The Assyrians cut off all the hair from their captives for three reasons: humiliation, sanitation (especially while traveling under crude conditions to Assyria), and separation (if any slaves escaped while being moved from their homeland, they could not blend in with other peoples since their baldness would give them away; thus they usually were quickly recaptured, punished, and returned to their captors.)" (Ludlow, p. 145.)