Bid is up 40% from late May when talks between the companies were first reported.
The company said the deal brings its valuation to $1 billion.
Two hedge funds have declined to participate in the compensation of the former retailers' U.S. workers.
The deal will "create a competitive automotive supplier which is extremely well placed among the global Top Ten.”
In a $6 billion deal, the private-equity firm will replace KKR as the majority owner of Sedgwick, the largest U.S. insurance claims service provider.
For Tipalti's new finance chief, successful experiences in consulting and private equity led to her entry into one of today's most dynamic industries.
Democratic lawmakers have asked KKR and Bain Capital whether they had "a deliberate policy" to load the defunct retailer with debt.
The $46-per-share deal “is the latest in a spate of mergers and acquisitions among physician networks."
"Companies that help simplify and manage essential infrastructure for their enterprise customers play an increasingly important role," said a KKR executive.
The business has been in decline as consumers turn to natural fats such as butter but KKR believes it has "a firm foundation for future growth."
The combination of American Medical Response and KKR's Air Medical will transport more than five million patients per year.
The KKR-controlled company will pay $2.8 billion for a market leader in online health with 71.7 million unique monthly visitors.
