Azaylia, who was born in August, last year, was diagnosed with leukaemia when she was eight weeks old and a fundraiser to try and save her life had reached more than £1.5m.
She had recently suffered a seizure, and her palliative care team told the family to “make the most of this weekend” as her body was “shutting down”.
Sharing a picture of himself holding his daughter’s hand, Cain wrote: “My heart is shattered.
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“You gave my life a purpose, my face a smile, filled my heart with love and my soul with pride.
“I can’t put into words my sadness and pain, there aren’t words to describe it.
“I love you with every ounce of my body and I miss you so much already.
“You made me a better man. You made me the person I’ve always wanted to be. You’ve taught me more in your 8 months than I’ve learned my whole life.
“I don’t know why you’ve been taken away from me and my heart aches in your absence.
“But I promise you that your name with live on.
“I will carry your name across every ocean, along every road, to the top of every mountain and across the whole globe.
“Your name will live, your legacy will be remembered and I will seek peace in living as your vessel to continue inspiring, giving strength, spreading joy and saving lives in your name, with your spirit, as your father until we meet again.”
Cain added that she had “saved” his life and given him a “second chance”, before ending the post with lyrics from the song I Wan’na Be like You from Disney’s The Jungle Book.
The 30-year-old reality TV star and Azaylia’s mother Safiyya Vorajee had been raising money for her to get specialist treatment in Singapore.
And despite a successful bone marrow transplant, her leukaemia had rapidly multiplied and Azaylia had gone into a full relapse.
Doctors said chemotherapy would not be enough to save her.
In November, the couple made a plea for people to register as a stem cell donor after being told their daughter required a transplant, which led to 41,000 people applying within 48 hours.