I searched birth/baptisms 1839 + - 5 years in Laois on www.rootsireland.ie/ for Maria Delaney with parents Edward Delaney & Mary Egan but got no matches.
I then searched for Ellen instead of Emily by mistake but got a match: 1842!
The parish isn't Mountrath but Raheen RC.
Could this have been a sister?
I then searched with no child's first name but with same parents' names 1839 + - 20 yeas and got 4 matches: Richard 1839 Margaret 1844 Ellen (same one as above) and John 1850.
Could your girl have been Ellen or Margaret do you think?