She said she had her work cut out for her with two swans; "I can deal with half a dozen of them, but no more."
Mrs Durrant said if there was a large amount of injured birds, the council needed to step in.
She advised that anyone who found a bird affected by the diesel should wrap it up and wash it gently with hot water and detergent.
Mrs Durrant said to release birds away from Lake Pupuke.
A Waitemata District Health Board spokesman said the diesel was believed to have originated from a small leak in a diesel storage tank on the North Shore Hospital site used to power hospital generators in the event of an electricity outage.
The leak - which had been contained - coincided with torrential rain which overwhelmed the hospital's stormwater filtration systems.
As a result of the storm, the diesel washed straight into the lake, overwhelming the containment mechanisms. Without the heavy rain, the stormwater filters would have prevented the diesel doing so.
The exact amount of diesel that had entered Lake Pupuke was unknown but it was believed to be relatively small.
However, he said even a small amount of diesel could spread over a large water surface area.
The DHB was working closely and with urgency with Auckland Council to remove a diesel slick and smell from a section of the lake.
Ongoing actions included the erection of booms across the stormwater entry point to the lake to contain the spread of diesel where possible and the erection of signage.
Notifications would also be sent out to key groups of lake-users and nearby residents and engagement would be ongoing with the Department of Conservation and Bird and ecology advisors to manage impacts.
Based on current evaporation and dispersion rates, the DHB had been advised the visible signs of diesel on top of the water should disappear over the next few days.
Auckland Regional Public Health Service said people should avoid contact with lake water in the affected area, including boating, fishing and swimming.
If contact had occurred, people were instructed to wash the affected area with soap and water.
They also advised that medical attention be sought if anyone had contact with the lake water and had developed any eye, respiratory or skin irritations.
Pets are to be kept away from the water and weed.
An Auckland Council spokesman said the Waitemata District Health Board was working with relevant groups including Department of Conservation following concerns for birdlife at Lake Pupuke.