He cannot pray Taraaweeh in the mosque because he is busy looking after his daughter
Taraaweeh prayer is Sunnah for which the one who does it will be rewarded, but the one who does not do it will not be punished. So there is nothing wrong with your not praying Taraaweeh with the congregation. Allaah knows the soundness of your intention and you may be rewarded even though you do not attend these naafil prayers, especially since you are not being negligent by not praying it. Indeed sometimes a person may be kept busy with something that is more important, such as one who takes care of an elderly mother or orphan children who need to be looked after at such times. A person may be rewarded more if he is distracted from it by a good deed that needs to be done at that time.
There is nothing wrong with praying Taraaweeh at home if you are unable to go to the mosque.
It is not necessary to pray a specific number of rak’ahs, rather you should pray as many as you can manage. The Sunnah is to pray eleven rak’ahs – see Question no. 9036. If you pray two rak’ahs that is better than not praying. We ask Allaah to compensate you with good for what you have missed and to record for you a reward that is greater than your praying Taraaweeh.
Your letting your wife go and visit her mother and your taking care of your daughter are two great deeds for which you will be rewarded in sha Allaah. See the answer to question no. 37742.
But we would like to draw your attention to two important matters:
2 – The ruling on staying in a kaafir country.
And Allaah knows best.