Her birth is more tricky. There are lots of Elsie Cook's in the birth indexes (did you know you can search the full B M & D indexes free on Ancestry at present?)
but JUST looking at 1896 and 1897, and JUST noting those that are plain Elsie without a middle name, and JUST those in the 7a area - there are these to choose from:
1896 Jan/Feb/Mar Leicester area ref 7a 273
1896 A/M/J Glanford B area ref 7a 692
1896 OND Glanford B area 7a 695
1897 JAS Grantham 7a 474