Hi, not a great deal you can do to quiet down your differential other than change the oil( which if it's whining means that the bearings have already sustained damage.
Unless you remove the gearbox from the car and strip then rebuild it. The cheapet and fastest way would be to source either a second hand unit from the likes of Ebay, breakers yard etc or purchace a recon unit (maybe on an exchange basis to further reduce costs)
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Try fixfault.com to fix cars.