"We call it the Government's gift, this leaky home, it's the gift that keeps on giving," she told the Herald.
The money they have to pay to fix a mistake that's not their own should have gone to caring for their three children, Busch says.
Both parents are former school teachers, but when Rheegan was born, Busch became a full-time carer, sleeping next to her daughter every night.
She's not had a full night's sleep in 10 years.
Earlier this year, Barry was forced to quit his job due to severe hearing loss and health difficulties, ending a career of almost 30 years.
The family receives financial help from the Government and disability organisations, but Busch says life's not a walk in the park.
The biggest task for the family will be moving out of their Titirangi home while it's reclad.
In their home they have safety glass, high gates, areas like the kitchen are barricaded and there are double locks on all the windows.
They have found a house to rent in Rodney through CCS Disability Action, but will have to spend thousands on modifications to keep Rhee safe.
"We have to modify - there's 100 ways that Rhee could die. I would hate for us to be in the position where we've come this far and we can't do it, because it's just not worth losing my baby girl," Busch said.
Government and council will provide half of the recladding costs under the Financial Assistance Package for eligible owners of leaky homes.
But there will be no compensation for the amount they pay on rent, or the mortgage payments which will increase when they borrow to pay for the other half of the recladding costs.
After a long fight, it's left the family feeling "dejected and hopeless", Busch says.
Through a Givealittle page, 383 donors have so far helped the family raise $56,000.
The company doing the work, Forme Reclad, has been "Herculean amazing" - with the owner helping the family with paperwork and doing little things that really make a difference.
"People have been humbling and inspiring," Busch says.
But Rhee's parents worry that if they manage to get a bank loan to reclad their house, the added mortgage pressure will mean they can't afford to keep it.
In addition, the move to Rodney will be a new source of anxiety for Rhee.
"The thought of moving her to a new place she doesn't know, with new sounds ... are we going to get back into the self-harming that we spent thousands of hours trying to lessen?"
The distance between Rodney and Titirangi means they will lose Rhee's carers, women who've worked with her for years.
"It takes a very long time to train someone to have Rhee without me," Busch said.
"If she freaks out because she hears a dog barking, and she's pinching you because she thinks she's fighting for her life, you have to love her enough to stay being hurt until you can get her somewhere safe - so it takes a pretty special person to look after her.
"Her carers have their own families. They're not going to travel two hours each day - and when we get back they may have taken other jobs, so it has a bit of a ripple effect, this Government's gift."
This means that Busch and Barry will lose the time they have to focus on other things important to them, like their 14-year-old son and Rhee's twin sister.
"I just hope that we can get the modifications that we need in the rental, we can live there without scarring my daughter, then we can get home and try to get back to our normal."
The family bought the monolithic cladding home for $600,000 10 years ago when both parents were working as full-time teachers.
The house was consented in 2005. The family had an architect inspect it, a building report was written and moisture testing was done.
The property's cladding also had a 15-year warranty, which has not been honoured.
"We didn't just jump in, we thought this would be our forever home," Busch said.
The leaky housing issue:
Leaky homes emerged as an issue in the 1990s, after a change to the Building Act (1991) meant building controls became more self-regulated.
As a result, weather tightness issues emerged in many buildings constructed between 1994 and 2004. The impact of changes to the Act were exacerbated by other factors - including the dropping of the apprenticeship training system, the popularity of Mediterranean-style (stucco) buildings and a building boom.
According to the Ministry of Business, Innovation and Employment (MBIE), a lack of proper supervision by local authorities and approvals that did not meet previous more rigorous standards was also a factor.
The total cost of leaky buildings in New Zealand was estimated to be about $11.3 billion for 42,000 buildings.
In Auckland, where the number of properties with open claims was 3139 at August 31, the council forecast claim payments of $487 million between 2011 and 2022.
As at August 31 2016, MBIE had received 7350 Weathertight Home Resolution Service (WHRS) claims lodged for 12,753 properties.
It is unknown how much leaky homes have cost homeowners. Between 2011 and July this year the Government's Financial Assistance Package scheme saw the Government and local authorities each contributing 25 per cent of the agreed repair costs. This means homeowners still have to foot 50 per cent of the bill.