When Lizzy and the Gardiners go to Derbyshire, we’re told “The children, two girls of six and eight years old, and two younger boys, were to be left under the particular care of their cousin Jane, who was the general favourite, and whose steady sense and sweetness of temper exactly adapted her for attending to them in every way—teaching them, playing with them, and loving them.”
Given that the Gardiners seem to have met after Mr. Gardiner relocated to London, it's likely the children were all born in Gracechurch Street.