In the October to December 1910 birth index Irene Bagnall. Ashton Under Lyne district, Lancashire, volume 8d, page 461.

In the 1911 census 133 Greenside Lane, Droylsden, Manchester
Head Herbert Bagnall, age 26, maker-up cloth trade, working for a dye works, born Hulme, Manchester.
Wife Rose Bagnall, age 25, married 5 years, 3 children born alive all still living, born Liverpool.
Son Edward Bagnall, age 5, born Hulme, Manchester.
Daughter Elsie Bagnall, age 3, born Beswick, Manchester.
Daughter Irene Bagnall, age 6 months, born Droylsden.

It would seem Irene died before her father (September 25 1915) as she is not listed in hiis pension records.

Findagrave index Irene Bagnall ndied September 21 1912, buried Droylsden Cemetery, Droylesden, Tameside, Manchester

Findagrave.com
Irene Bagnall
Birth 1910
Death 21 Sep 1912 (aged 1–2)
Burial Droylsden Cemetery
Droylsden, Metropolitan Borough of Tameside, Greater Manchester, England
Plot Section P, Grave 390
Memorial ID
219800680 Age: 2, Death date listed may be the interment date