The baby, who has not been named, died with her brothers Adhyan, nine, and Amaan Nazim, seven.
Two other members of the family – the children's grandmother Shabina Begum, 54, and her daughter Anum Parvaiz, 20, the children's aunt – were also killed in the fire.
The fire broke out in the family home at Wake Road, Sharrow, last night (Sunday, April 17). Around 20 firefighters tackled the fire shortly after midnight.
He described the fire as 'a tragedy beyond imagination', adding that Shabina 'gave her life trying to save her grandchildren'.
The children's mother escaped the fire and their father, Nazim, was at work at the time.
Mr Kayani said: "The whole family is in a state of shock, devastated. The whole community is in a sombre mood. Friends, relatives, we're all in huge, huge shock. It's a tragedy beyond imagination.
"Just in a matter of minutes the whole family has been destroyed. The whole family has gone.
"You can well imagine what the children's parents are going through. I don't have the words to describe how the family is feeling, other than shell-shocked, devastated.
"We are a close-knit family, there are enough people around to support the family. The children were looking forward to school, and we've lost the whole family, just a wonderful family, highly respected."
Four fire engines were called to the house to tackle the blaze. Police and ambulance services also attended.
A fire brigade spokesman said: "An investigation has commenced, involving police and fire officers, to establish the cause of the fire."
A South Yorkshire Police spokesman said: "At this stage we are not able to confirm where in the house the fire broke out or if anyone else was injured."
Police officers and firefighters remained at the scene, he said.
A spokesman for South Yorkshire Fire and Rescue said details about the fire would become clearer later today.
He said: "Our investigators are arriving here at 9am, so we will know more later on this morning."