This page from my Hagia Sophia website shows the sources that were used in the creation of the website.
The garden is found next to the Church of the Nativity of the Virgin, at the end of the Corridor of the Maids of Honor. It links the Grand Palace to the Apartments of Their Majesties.
The greenhouses of the Alexandrinskii Summer Palace provide the plants for the Winter Garden.
One will note an adventurine vase in the garden, made out of one solid block, and a statue in marble, depicting Hercules and Vesta.