He led his wife and children in praying qiyaam al-layl and did not recite out loud

Firstly: 

Praying
Taraweeh at home is permissible and there is nothing wrong with it, but
offering the prayer in congregation in the mosque is better. 

The
scholars of the Standing Committee were asked: 

The month
of Ramadaan has come and it is time to pray Taraweeh. Should I go to the
mosque or pray in my house? I am not an imam, rather I am one of the
congregation, but I like to recite Qur’aan, and I prefer to recite rather
than to listen. If I pray at home, is there any sin on me? I am referring to
Taraweeh prayer only. 

They
replied: 

There is
no sin on you if you pray in your house because it is naafil. But praying
with the imam in the mosque is better, following the example of the Prophet
(peace and blessings of Allaah be upon him) and his companions (may
Allaah be pleased with them), because when the Prophet (peace and
blessings of Allaah be upon him) led his companions in prayer at night until
the end of the first third of the night, one of them said to him: We wish
that you would lead us in naafil prayer for the rest of the night. He said:
“Whoever prays qiyaam with the imam until he finishes, it is as if he spent
the whole night in prayer.” Narrated by Ahmad (5/159) and the authors of
al-Sunan with a hasan isnaad from the hadeeth of Abu Dharr (may Allaah
be pleased with him). End quote. 

Fataawa
al-Lajnah al-Daa’imah
(7/201-202). 

Secondly: 

The basic
principle concerning Taraweeh prayer is that it is one of the prayers in
which Qur’aan is to be recited out loud, because this is proven from the
Sahaabah (may Allaah be pleased with him) at the time of ‘Umar ibn
al-Khattaab (may Allaah be pleased with him), when Ubayy ibn Ka’b and others
would lead the people in prayer and recite at length. 

But
reciting out loud in prayers in which Qur’aan is recited out loud, and
reciting silently in prayers in which Qur’aan is recited silently, is
something that is recommended in prayer for the imam, but it is not
obligatory, as is the view of the majority of scholars among the Maalikis,
Shaafa’is and Hanbalis. 

It says in
al-Mawsoo’ah al-Fiqhiyyah (16/188): 

The
majority of fuqaha’ are of the view that reciting out loud in prayers in
which Qur’aan is recited out loud, and reciting silently in prayers in which
Qur’aan is recited silently, is one of the Sunnahs of prayer. The Hanafis
are of the view that it is obligatory to recite out loud in prayers in which
Qur’aan is recited out loud, and to recite silently in prayers in which
Qur’aan is recited silently. End quote. 

Shaykh Ibn
‘Uthaymeen (may Allaah have mercy on him) said in Noor ‘ala
al-Darb (al-Salaah/218): 

Reciting
out loud in prayers in which Qur’aan is recited out loud is not obligatory,
rather it is preferable. If a person recites silently in a prayer in which
it is prescribed to recite out loud, the prayer is not invalid, because the
Prophet (peace and blessings of Allaah be upon him) said: “There is no
prayer for the one who does not recite the Essence of the Qur’aan (i.e.,
al-Faatihah).” And there is no stipulation that this recitation should be
done out loud or silently. So if a person recites what he is required to
recite, whether it is silent or out loud, he has done what is required. But
it is better to follow the Sunnah in prayers in which Qur’aan is recited out
loud, such as the first two rak’ahs of Maghrib and ‘Isha, and Fajr prayer,
and Jumu’ah and the Eid prayers, and prayers for rain (istisqa’) and
Taraaweeh prayer and so on, where it is customary. If a person deliberately
does not recite out loud when he is leading the prayer, his prayer is still
valid, but it is lacking. As for the person who is praying alone, if he
offers a prayer in which Qur’aan is recited out loud then he has the choice
between reciting out loud or silently, so he should decide which will make
him focus better, and do that. End quote. 

It says in
Fataawa al-Lajnah al-Daa’imah li’l-Ifta’ (6/392): 

It is
proven that the Prophet (peace and blessings of Allaah be upon him)
used to recite Qur’aan out loud in the two rak’ahs of Fajr, and in the first
two rak’ahs of Maghrib and ‘Isha’ prayer, so reciting out loud in these
prayers is Sunnah, and it is prescribed for his ummah to follow his example
because Allaah says (interpretation of the meaning): “Indeed in the
Messenger of Allaah (Muhammad صلى الله عليه
وسلم) you have a good example to follow for him who hopes for (the
Meeting with) Allaah and the Last Day, and remembers Allaah much” [al-Ahzaab
33:21].  And it is proven that the Prophet (peace and blessings of
Allaah be upon him) said: “Pray as you have seen me praying.” 

So if he
recites silently in prayers where Qur’aan should be recited out loud, he is
forsaking the Sunnah, but his prayer is not invalidated by that. End quote. 

To sum up:
your prayer is valid and you do not have to do anything else. 

And Allaah
knows best.