The eight-year-old daughter who survived did not have explosives strapped to her, but was thrown three m (10 ft) into the air by the blast and was receiving intensive care in hospital, police said. Six civilians and four officers were wounded.
The attacks killed 25 people, 13 of whom were the suicide bombers, and injured dozens more.
Police said they killed four suspected members of JAD in Cianjur, West Java Province, and arrested two others, the BBC reported.
The coordinated bombings on Sunday, followed on Monday by a suicide attack by another family on police headquarters in Surabaya, have horrified Indonesians who typically see their Muslim-majority country as diverse and tolerant.
The attack came just hours after police said the family that carried out the church bombings included girls aged 8 and 12.
The attacks are believed to have been instigated by radical Indonesian group, the Jemaah Ansharut Daulah network, which has pledged allegiance to Islamic State.
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ISIS, from which the recent attackers reportedly took their cues, has always understood the power of using provocative violence, Barton said.
All told, 26 people died in the two days of attacks including a total of 13 militants and their children. Jemaah Islamiyah, the al-Qaida affiliated network responsible for the Bali attacks, was obliterated by a sustained crackdown on militants by Indonesia's counterterrorism police with US and Australian support.
Democratic Party deputy secretary-general Rachland Nashidik said that while it was important to revise the Terrorism Law, the President should "pay attention to organizational reform and replace officials who have failed in carrying out their duties to protect the freedom and security of citizens".
Indonesia's latest wave of terrorist bombings, the worst since 2009, have killed at least 14 people.
Later in the day, police said another family attempted to carry out an attack the Surabaya suburb of Sidoarjo. Its leaders were killed in police raids and hundreds of militants were arrested. "Now if you go to the airport, there's a tight security check, and even if you go to some housing areas and malls they are also conducting security checks", Michelle said.
At Surabaya Centre Pentecostal Church, Futrianto then drove his auto onto the church grounds and detonated explosives. The mother and her 17-year-old daughter were killed when a bomb handled by the father exploded prematurely.