Esther Kidemba died after apparently jumping from a building during a drill at the private Strathmore University in the capital, Nairobi.
Gunshots were fired during the drill, causing panic on the campus.
Militant Islamist group al-Shabab killed some 150 people in an attack on Garissa University College in April.
Strathmore University Vice-Chancellor John Odhiambo said he offered his deepest condolences to the family of Ms Kidemba, who was a staff member, and an "unreserved apology to every student, parent, family, colleague and stakeholder for the unfortunate outcome of the security drill".
Strathmore would pay for the medical expenses of 30 students and staff injured during the drill, and would arrange post-traumatic counselling, he added.
The drill had been carried out by the university's security team in co-ordination with local police to assess how they would deal with any attack, a statement by the university said.
On Monday, Nairobi police chief Japheth Koome said all the proper procedures had been followed for the mock security exercise, Reuters news agency reports.
But in his reaction, police chief Joseph Boinett said: "This must not happen again."
Kenya's security forces were on high alert to deal with threats, and drills should be conducted only with the authorisation of the "highest security office in the country", he added.
April's day-long assault on Garissa University College in north-eastern Kenya was the deadliest by al-Shabab in the East African state.
In 2013, at least 67 people were killed in an attack by the al-Qaeda-linked group, which is headquartered in neighbouring Somalia, on the upmarket Westgate shopping centre in Nairobi.
Al-Shabab says it is opposed to the presence of Kenyan troops in Somalia.
The troops are part of an African Union (AU) force helping the weak Somali government fight the militants who want to establish Islamic rule in the country.
Kenya's police chief has warned universities not to carry out security drills without his approval following the death of a woman on Monday.