Likely but she was in and out of the asylum at least twice in 1910Seems then that she would have had her second son whilst a patient. He was registered Dec qtr 1910.
There's a death for a William Steeles of the right age registered Bham N, Jun qtr 1931. In 1911 he's in Birmingham.
The eldest son of William and Emily seems to have been adopted by John and Annie Gilbert, the other son dying young.