Her birth is not easy to be sure of but the following core information applies to her as the mother of John McDermott:

1. The mother’s maiden name of Sharp on the birth index of all her children.
2. The 1911 census giving her 5 years of marriage.
3. The marriage in 1906 in North Bierley district, which is now in parts of Bradford, Keighley, Wharfedale, Spen Valley, and Leeds South registration districts and at least adjoins Bramley sub-district.
4. I clearly found her with her family in 1939 and there said to be born on 10 Feb 1888.
5. Her first five children were born in the Bramley district.

Accordingly this 1901 census entry in Bramley looks very like hers at 2, Billingbank Terrace, Bramley, Yorkshire (West Riding):

First name(s)  Last name  Relationship  Marital status  Sex  Age  Birth year  Occupation  Birth place
John         Sharp  Head  M  M  38  1863  Hydraulic engineers draughtsman  Mold, Flints, Wales
Elizabeth A  Sharp  Wife  M  F  39  1862  -                                Leeds, Yorkshire
Elsie        Sharp  Daur  -  F  13  1888  -                                Hunslet, Yorkshire
Leslie       Sharp  Son   -  M   7  1894  -                                Hunslet, Yorkshire
William      Sharp  Fthr  W  M  79  1822  Retired foreman mechanic         Bowling, Yorkshire

Ref:
Archive reference  RG13
Piece number       4206
Folio              111
Page               29
Combined:          1901 ref: RG13, piece 4206. folio 111, p. 29
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Then a baptism has been found at Hunslett where she was reported, above, to have been born in the 1901 census.  This included her birth date but said that it was on the 16th February 1888, not as on the 10th as clearly written in the 1939 register.  Otherwise the information corroborates that in the 1901 census.  This transcript is by me direct from the image of the baptism register page:

“                               Page 7
“BAPTISMS solemnized in the parish of Hunslet in the
“County of York in the year one thousand
“eight hundred and Eighty Eight
“. . . .
“When born:           1888 16th Feby.
“When baptized:       1988 April 15th
“No:                  53
“Child’s Forename:    Elsie Dr of
“Parents’ Forenames:  John & Elizabeth Ann
“Parents’ Surname:    Sharp
“Abode:               Longwood Grove
“Profession:          Draughtsman
“Celebrant:           J S StJohn Corlott(?)

In fact I have taken her above birth date of the 16th as I believe there was less room for error so close in time to the event.
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And the above birth is corroborated by the Birth registration index:

First name(s)         Elsie
Last name             Sharp
Birth quarter         1
Birth year            1888
Mother's maiden name  Thorp
District              Hunslet
County                Yorkshire
Country               England
Volume                9B
Page                  290
Record set            England & Wales Births 1837-2006
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In conclusion about Elsie’s birth and parentage, I suspect that the only way to be more sure of this is to get hold of a copy of her marriage certificate in the hope that the name and occupation of her father will be sufficient to identify her parents.
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Her death index:

First name(s)  Elsie
Last name      McDermott
Gender         Female
Birth year     1889
Age            71
Death quarter  1
Death year     1960
District       Leeds
County         Yorkshire
Volume         2C
Page           241
Country        England
Record set     England & Wales Deaths 1837-2007
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Their marriage index:

First name(s)     Joseph Michael
Last name         McDermott
Marriage quarter  4
Marriage year     1906
MarriageFinder    Joseph Michael McDermott married one of these people
                  Elsie Sharp, Agnes Ethel Fisher
District          North Bierley
County            Yorkshire
Country           England
Volume            9B
Page number       436
Record set        England & Wales Marriages 1837-2005
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