"I can confirm that I did direct Immigration New Zealand to grant a nine-month visa for Jendah," Ms Kaye said.
When asked if she would support an application to allow Jendah to stay permanently, Ms Kaye said she would "consider any future application on its merits".
"I generally do not comment on the detail of a case while it is being processed or future applications before I have the opportunity to consider the merits of the case," she said.
Speaking from hospital in Cambodia, where she has been treated for dengue fever, Mrs Dyer said she hoped to be coming home next week with Jendah.
Mrs Dyer has been living and working in Cambodia with her husband as Asian Outreach New Zealand aid workers, with their two other children aged 11 and 12.
However, Mr Dyer had returned to New Zealand last month with their children when his father passed away.
Mrs Dyer said Jendah's passport and visa application had been sent to Immigration's Bangkok branch for processing.
"As soon as we get her passport we will be able to fly home, and by the miracle of God it will be Monday."
"Jendah's really excited, she doesn't have the words to express things, but you can just see it from her reaction when we talk about going home."
This would be Jendah's second trip to New Zealand, the first was a two-month visit in 2009 under the "guardianship" of Mr and Mrs Dyer.
However an Immigration spokeswoman said Jendah's health became an issue because her adoption did not meet applicable legal requirements and a subsequent visa application lodged in 2011 was declined.
Mrs Dyer said Jendah misses her elder sister and has been speaking with her over the phone frequently since she left.
"Jendah just can't wait to see her sister and be with the rest of the family again," Mrs Dyer said.
Mrs Dyer said she planned to apply again for citizenship or at least obtain permanent residency for Jendah, so that the family could remain together in New Zealand.
"I still believe Jendah deserve citizenship because we, her parents, are Kiwis and New Zealand is where her family and extended family are."
Jendah Dyer
2009: First visit to NZ under "guardianship" of Grahame and Kathryn Dyer.
2011: Adopted by Mr and Mrs Dyer under Cambodian law.
2011: Second visa application lodged.
2012: Application declined because she did not meet Immigration health requirements. Citizenship application also declined because NZ did not recognise Cambodian adoptions prior to Jan 1, 2012.
2013: Associate Immigration Minister directs INZ to issue a 9-month visa.