principle he, his wife Wilhelmina, and their three eldest children should all feature on the 1881 census as well (since they didn't all pop into existence fully-formed).
So given that in 1891 we have
James Thornhill Head 35 Cashier at Ironworks born Shipley, Derbyshire
Wilhelmina Wife 32
William P Son 14 Auctioneers assistant
Ethel M Dau 12
Wilhelmina Dau 11
J Arnold Son 10
in 1881 we should have
James Thornhill Head 25
Wilhelmina Wife 22
William P Son 4
Ethel M Dau 2
Wilhelmina Dau 1
J Arnold Son ? months
Perhaps not quite like that and perhaps not with those exact ages (since people could often be a bit vague), but there should be something
What we actually find is NO Wilhelmina Thornhill, and no William, Ethel or J Arnold of around the right age except for a few who are listed with parents that are very definitely not James or Wilhelmina.
And the only James Thornhill of the right age and place of birth is the one I listed in my earlier post.
I am absolutely not suggesting that he is the same James Thornhill who was listed as Violet's father in 1891. That's the sort of huge leap of logic that you need to be very careful to avoid. But it is one possibility that should be kept at the back of your mind.
For example, it's possible that James and Wilhelmina were not actually married at the time of the 1881 census, that the eldest few children initially took Wilhelmina's name (hence why none of them show up under "Thornhill" on the 1881 census) and were living with her while James worked away from home as a miner for a few years to raise money. Then that somewhere between 1881 and 1891 James had made enough money to marry Wilhelmina, the children all took on the name of Thornhill, and he settled into his far more respectable career as a cashier at the ironworks.
All that's possible, but it could just be a romantic fiction built up on a couple of shreds of evidence. It's just as possible, perhaps even more so, that the census enumerator was a bit deaf, or his handwriting very bad, so that they are all living happily together on the 1881 census but listed as Thornbush or Thorburn or Turnbull or something.
This is why your next step should be to get Violet's birth certificate. She was born in 1882, so only a year after the 1881 census. And she was registered with the surname Thornhill, so we can assume that her parents were married at that time.
From Violet's birth certificate, you will be able to find out (a) the address where she was born, so that you can go and check that out on the 1881 census and see if the family is living there (under whatever name) (b) Her mother's maiden name, so that you can look her up in the 1881 census under that name and see if there is anything.
This is roughly the conversation that bubblerock and I were having last night, but we were having it in shorthand rather than spelling out all the implications so it was probably a bit difficult to follow...