Her father Jack had been removing the fireplace when Aisling ran into the room and pulled it down on top of her, it is understood.
She lost her fight for life at Cork University Hospital where her life-support machine was turned off.
The accident happened at her home on Bere Island, off Cork's southwest coast.
Aisling was Jack and Anne Sullivan's only daughter.
The couple have four sons.
President McAleese had been due to visit the island for the first time over the weekend, but cancelled the trip following the tragedy.
She had been due to officially open the Bere Island Heritage Centre, which was built by Mr Sullivan.
The visit was called off as a mark of respect to the family.
"It is a terrible tragedy," an islander was quoted as saying, adding that their "hearts go out" to the family.
Mr Sullivan is one of the island's most successful and respected businessmen.
Aisling reposed at the family home over the weekend.
Her Requiem Mass will take place at noon tomorrow at St Michael's Church, Bere Island, followed by a private removal to the island's crematorium. Her grieving parents have asked that, in lieu of flowers, mourners donate to Barnardos Counselling Services.
The proposal to build the heritage centre had been many years in the planning before it became a reality in 2009.
Funding was received from the Department of the Islands and Cork County Council.
The centre has a craft shop and cafe with internet access, as well as an exhibition space, meeting room and conference facilities. The island has a population of just over 200.
The Bere Island country and western festival, which had been due to take place over the weekend, was also cancelled due to Aisling's death on Thursday.