Is it obligatory to sacrifice something to escape a painful accident, especially since no 'aqiqah was performed for a two-month-old daughter?

It is not obligatory for someone who has been saved from harm or has attained a desired outcome to slaughter an animal. Rather, it is recommended for them to perform a prostration of gratitude (sujud ash-shukr) upon receiving a blessing or repelling an affliction. If they wish to give charity or slaughter an animal as an expression of gratitude to Allah, there is no harm in that. The 'aqiqah for a girl is one sheep, and it is a confirmed Sunnah that can be performed at any time it becomes easy if its usual time has passed.