This is the 13418th most frequent Russian word.
найдя
"Having found."
The word 'найдя' is used here as a participle to indicate that the action of 'finding the ball' occurred before the main action of 'starting to play'.
Мальчик, найдя мяч, сразу начал играть.
The boy, having found the ball, immediately started playing.
Similarly, 'найдя' serves as a participle here to signify the completion of finding the key prior to opening the door.
Найдя ключ, женщина смогла открыть дверь.
Having found the key, the woman was able to open the door.
In this sentence, 'найдя' is used to mean that the act of finding the solution brought about the feeling of happiness.