Answer :
i think that in this context, ardent is most likely means : powerful
The sentence convey that the children rigorously waiting for the door to for a pretty long time.
As soon as the door opened, they did not even watch how they got out which cause them to almost fell out
Ardent means very enthusiastic or passionate, we can feel in this passage that the children’s feeling was very strong, is says that “they were about to fall out of the open door” these kids were kind of desperate until they saw their father, the best word to replace ardent in this case would be powerful based in the actual meaning of the word and the context were it is being used.