If you can afford better replace it with new IMO... that's what i did about 6 years ago when mine leaked again and it's still OK as previously i went for the rebuild kit and lasted just a bit more than 1 year... the gist is that if it's leaking at the shaft there is circular wear on it caused by the seal all over the years so a new seal will just keep "chewing" it and leak again
My pump is also leaking and probably it makes the whining noise as well when I turn the wheel. It is hard to really make sure that it is the pump but I have already checked everything and there is no any leak around the pump. If I take a look from the wheel arch it looks that the bottom the pump is oily and oil drops can be seems on the bottom of the hose connecting the pump to reservoir. As the oil there is already contaminated it is hard to judge if it is PS fluid of engine oil.
As above proposed by Fery probably I will order a new pump and replace it. Is there a make you can recommend? I have found on a German website (it is difficult to order now from the UK to EU) pumps from 60 to 200 pounds (Ridex).