Krystal Gamman first raised the issue around a lack of insulation with the property manager a year ago and even offered to use her community card to reduce the cost, the documents showed.
Their youngest child was hospitalised with respiratory issues two years ago and had pneumonia this winter.
The family complained the 1950s timber weatherboard cottage was cold in winter and there was a lot of condensation around the windows despite their best efforts to ventilate. The Gammans' request was ignored, the tribunal decision said.
In May this year, a contractor attempted to assess the property, but was unable to access the property because their daughter had a high fever and they had to rush her to the doctors.
A recent assessment of the property by an insulation expert revealed the cottage did not have the correct insulation - the ceiling insulation was insufficient and there was no underfloor insulation.
The Tenancy Tribunal was critical of the landlord waiting until two months prior to the deadline to carry out an assessment and said there was a strong public interest in deterring a landlord from ignoring its obligations.
"The Gammans' interests have not been served as they have been concerned about dampness in the house with young children, one of whom was hospitalised last year with respiratory issues, and is now in winter with suspected pneumonia," the order said.
Krystal Gamman told the Herald taking the property management company to the tribunal was a tough decision because they had a good history with them.
"It's about our children's health at the end of day and that's the reason we went ahead and did the dispute. It was never a personal matter it was just hey, this is two weeks past when you were meant to have this done."
The Gammans were happy with the outcome and are looking forward to the insulation being completed by the end of the week.
Wellington based firm Specialist Property Managers have also been given until August 20 by the Tenancy Tribunal to fix a leaking window and put in underfloor insulation at a flat in Berhampore that it manages on behalf of the owner.
The company came to an agreement with its tenants Ethan Helliwell, Felix Woodruff, and Matthew Loveman during the tribunal hearing, who were also awarded $520.44 in costs.
The maximum amount in damages that can be awarded by the tribunal is $4000.
These are the first two decisions to be released, but many more are likely.
The Ministry of Business, Innovation and Employment (MBIE) has encouraged tenants to discuss it with their landlord before applying to the Tenancy Tribunal. There is no grace period because landlords have had three years to prepare.
MBIE estimated 100,000 rental properties were still not insulated when the rules aimed at making homes healthier for Kiwi renters came into force at the beginning of July.
The Insulation Association says there has been a high demand for insulation and some landlords say this has caused delays in them meeting the July deadline.