Mrs Kivlehan, an Indian-born Irish citizen, was the sales director of a Next store in Sligo.
Husband Kevin (33) wants to know why she died, said family solicitor Roger Murray.
Her son, Dior, was born by C-section on September 21, 2010.
After complications following the birth, she was transferred to the Royal Victoria Hospital in Belfast four days later, where she died on September 29.
Mr Kivlehan's request for an inquest in the North has been refused by Belfast Coroner John Leckey and the family is now seeking an inquest in Sligo.
The case against the HSE is understood to centre around when medical treatment was administered.
"The family are unable to comment on the legal proceedings, as they are before the court," said Mr Murray.
"They hope, however to obtain answers to the serious questions they have about how Dhara's death occurred.
"In this regard, they are very concerned that an inquest be convened to inquire into the circumstances of her death. No inquest was held in Northern Ireland, where she died.
"Consequently, the family has corresponded with the Sligo Coroner, urging him to hold an inquest.
"An outcome on their request is expected shortly".
The sales director's death has left her husband and their families "completely devastated", said Mr Murray.
The HSE is expected to lodge a defence to the claim for alleged negligence and alleged breach of duty in the High Court in Dublin next Monday.
Family friends say the Kivlehan and the Sandhu families will travel to Dublin for the hearing.
"Dior is a beautiful boy and is the light of all their lives," said one friend yesterday.
"He was just eight days old when his mother died and the family want to be able to tell him how she passed away."
The couple first met in London, where Dhara was a fashion design student.
They moved to Friarstown, Dromahair, Co Sligo, in 2005 when she landed her job with Next.
"Kevin will never get over her death," said another friend.
"He knows he cannot speak until the court proceedings are over, but it is a dreadful loss and he really just wants to know what happened."