I agree you need the birth certificate, but I very much doubt it will name the Duke of Rutland! Given that the birth was registered as Mills with mmn Williams it will probably show Queenie as daughter of Agnes Mills née Williams and either be silent as to paternity or contain the name of her late husband as though he were still alive. One of my ancestors claimed a child born in 1837 was born to herself and her husband - problem was, her husband had in fact died 4 years earlier!
From the newspapers, the Duke seems to have been busy carrying out public duties, including in his role as Lord Lieutenant of Leicestershire, in the autumn of 1916 when Queenie would have been conceived (born 20 July 1917).
I wonder how a link has been established (if indeed it has) - perhaps by documented acknowledgement at the time, or by modern DNA?
Interestingly there seems to have been acknowledgment that the Duke’s own youngest daughter (as recognised by law), Lady Diana Manners b 1892, was in fact fathered by another man, Henry/Harry Cust.
