My dear Vanya,
I am not sure but I don't think it would be the Duchess of Kent. Although Edward may have been fond of his grandmother, Victoria did not really have any connection to Sandringham unless she presented the portrait to Edward and Alexandra as a gift.
I thought it might be the Queen Mother because it resembles the cover photograph on the Shawcross biography. Her majesty is wearing a dark dress which is slightky off her shoulders and she is wearing the Indian tiara. In addition, it appears that the portrait contains some ermine robes which to me signifies that the subject has been crowned. But then upon closer examination, the style of the portrait appears to be too old to be the Queen Mother. Could it be a member of the Danish royal family and thus a relative of Queen Alexandra?