Katherine was sent to Christchurch from Southland for treatment, where she caught a fungal infection, a Givealittle page for the Thompson family said.
Gravely ill, the 11-year-old returned to Southland and was for a time receiving palliative care in Invercargill Hospital. She died on June 26.
"We cannot comprehend the pain and grief you are facing during this time," a post on the Givealittle page said.
"Our thoughts and prayers of support, love and strength are continuing for Frances, Amelia, Jack and family.
"Life is not measured by the breaths we take, but the moments that take our breath away. Rest peacefully beautiful girl."
More than $14,000 had been donated by around 300 people in the 13 days since the Givealittle page was created.