a few more records to keep you busy.
Elizabeth Audley was living with Peter and Jane Keever in 1870 in New York City. Jane's maiden name was Audley. I couldn't tell if she was Elizabeth's sister, cousin, or whatever. Maybe her full death record will list her parents' names. But not a real good chance in that time frame.
Jane died 5 July 1875, Manhattan, age 32, marital status unknown [according to the transcript]
"New York, New York City Municipal Deaths, 1795-1949,"
https://familysearch.org/ark:/61903/1:1:2WYY-TDN------
Jane's headstone reads:
Erected by Mary Keever in memory of her beloved parents
Peter Keever died April 8? 1884
Jane Keever died July ?, 1875
her infant brother E J Keever
https://www.findagrave.com/memorial/196491737/jane-keever------
Here is the transcript of the death record of Peter and Jane (Audley) Keever's child, Mary, which gives Jane's maiden name:
New York, New York City Municipal Deaths
Name:, Mary Keever
Event Type:, Death
Event Date:, 09 Aug 1930
Event Place:, Bronx, New York, New York, United States
Gender:, Female
Age:, 59
Marital Status:, Single
Race:, White
Occupation:, Laundress
Birthplace:, New York
Burial Date:, 11 Aug 1930
Cemetery:, Calvary
Father's Name:, Peter Keever
Father's Birthplace:, Ireland
Mother's Name:, Jame Audley
Mother's Birthplace:, Ireland
https://familysearch.org/ark:/61903/1:1:2WG8-YNB=========
If your John Joseph Audley is the John Audley who was naturalized 3 November 1888, Circuit Court, Buchanan County, Missouri
Volume A, page 438
country of birth or allegiance: England
then chasing down his petition for naturalization might tell you his county of birth, there's about a 50-50 chance.
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You said that John Joseph Audley may have been born in Belfast. For what it's worth, I found this for an Elizabeth Audley, but with a father named Pat:
baptized 21 September 1848
St Patrick's, Belfast
Elizabeth Audly, daughter of Pat Audly and Elizabeth Gorton. sponsor: Catherine O'Donnell
I wouldn't put a lot of stock in it, for being the right Elizabeth. Also I found no Jane or John in the baptisms in Belfast, according to ancestry.com's reading and indexing of the names.
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If you can tie your Audley family to this one, also in Kansas City, Missouri, then you will know where yours came from.
I found this first:
1910 Census Ward 5, Kansas City, Missouri
at 4013 Belleview Avenue
Thomas Burke, 43, head, born in Ireland, arrived 1888, naturalized
Maria Burke, 37, Ireland, married 1x, for 13 years, 6 children, 6 alive, arrived 1890
Ellen, 12, Missouri
Mary, 9, Catherine 7, John W 5, Thomas M 3, Joseph P 1, all born in Missouri
John Audley, 27, brother in law, born in Ireland
Margaret Audley, 25, sister in law, born in Ireland
Richard Halloran, 22, Ireland, boarder
https://www.familysearch.org/ark:/61903/1:1:M2Y8-F1S------
When John and Margaret, brother and sister, arrived, they were headed to their sister Bridget's:
Arriving New York, 1 May 1902, from Queenstown, on the Lucania
John Audley, 19, single, laborer,
Maggie Audley, 17, single, servant
both: last residence: Headford, destination: sister Bridget Audley, 1016 Centre St, Kansas City, Missouri
Headford is in Galway
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Bridget, John, and Maggie are all at home in Galway in 1901:
1901 Census of Ireland
house 3 in Pollacullaire (Headford, Galway)
Audley, Thomas, 55, Male, Head of Family, General Labourer, Cannot read or write, speaks Irish and English, Married, -
Audley, Katie, 50, Female, Wife, House Keeper, Cannot read or write, speaks Irish and English, Married, -
Audley, Bridget, 24, Female, Daughter, Read and write, speaks Irish and English, Not Married, -
Audley, John, 19, Male, Son, General Labourer, Read and write, speaks Irish and English, Not Married, -
Audley, Maggie, 17, Female, Daughter, Read and write, speaks Irish and English, Not Married, -
Audley, Katie, 14, Female, Daughter, Scholar, Read and write, Not Married, -
Audley, Nora, 7, Female, Daughter, Scholar, Read and write, Not Married
all Roman Catholic, all born in Galway
http://www.census.nationalarchives.ie/pages/1901/Galway/Headford/Pollacullaire/1399892/