RootsChat.Com

There’s so many women with that name in almost every state. And we don’t know if King is her birth name or a married name.
Do you have any other names associated with her at all? 

Judith

This is a VERY long shot and I can't see how it could be proved that this is the correct person.

New South Wales, Australia, Unassisted Immigrant Passenger Lists
Arrival, Sydney 9 July, 1866
Ship Ellen Southard
The list shown are all crew, with the position held by each person next to their name and among them is:
William Donnelly, 22, OS, Nationality given as Great Britain (OS = Rank of Ordinary Seaman which implies that he had had little or no previous experience as a sailor)
Ship came from Tome which is in Chile, while newspaper reports say it left Talcahuano, Chile which is about 30k  away from Tome, having left Chile on 28 March.  The ship appears to have been carrying wheat and flour.
There are a number of mentions in Trove but all with similar information although one mention suggests that it is an American ship
From the Sydney Mail, Sat 21 Jul 1866
ELLEN SOUTHARD, American ship, 828 tons, Howe, at Walker's Wharf, agents: Wilkinson Brothers, and Co.
Wikipedia says: Ellen Southard plied international trade routes from her homeport in Bath, Maine, with visits documented in ports as far away as Sydney, Australia.
Trove has the Ellen Southard in Sydney in 1865 and she leaves Sydney at about the end of June 1865 bound for New Caledonia. I don't know where she went between bein in New Caledonia and Chile - maybe back to the US?

So perhaps this is William???

Judith

If you know her name and she came as an immigrant you may find her name on this site which has assisted immigrants from that time. 

If you know she is deceased Rootschat researchers may be able to find her on electoral rolls etc However you may not feel comfortable in disclosing her name and she could still be living.

Judith

What a sad and distressing story to be reported publicly in the newspaper, with the additional mention of the brother.
So it seems likely that the 1869 death is the correct one, although the father’s name doesn’t match that given by the OP.

Judith

Find-a-Grave has this burial, but I can't see it on the NSW BMD index so can't verify any other details to see if this is the correct person.

William Donnelly
BIRTH 1841
DEATH 1 Jun 1869 (aged 27–28)
BURIAL
Gladesville Asylum Cemetery, Gladesville, Hunter's Hill Municipality, New South Wales, Australia
PLOT 335
MEMORIAL ID 191590699 · View Source

Quite a lot about the hospital and its cemetery is available on Google, but couldn't see anything specific about William although many of these burials were, apparently, in unmarked graves.

Judith