How can one perform the Tarawih prayer at home while caring for a daughter, given the wife's absence?

Tarawih prayer is a Sunnah (definition) for which one is rewarded for performing it and not punished for neglecting it. There is no harm in not praying it in congregation. If Allah knows the sincerity of one's intention, a person may be rewarded even if they do not attend supererogatory prayers, especially if they do not habitually neglect them. A person might be occupied with something greater, such as caring for an incapacitated mother or orphans, and they may be rewarded more for that. There is no harm in praying it at home if one is unable to go to the mosque, and there is no specific number of rak'ahs required; rather, a Muslim prays according to their capacity and energy. The Sunnah is eleven rak'ahs, and two rak'ahs are better than none. A husband's permission for his wife to visit her mother and his care for his daughter are two great matters for which he will be rewarded.

The questioner is alerted to the impermissibility of a wife traveling alone without a mahram (definition), and the ruling on residing in non-Muslim lands.