Qiyaam al-Layl is one of the best of deeds and acts of worship, and the one who misses out on it should grieve and feel sorrow. But if a person is sincere in his intention and is resolved to get up, and takes measures to help him to do that, but then something prevents him, such as the crying or screaming of a child, then there is the hope that he will attain the reward of qiyaam, because of the hadeeth of Abu Moosa (may Allah be pleased with him), according to which the Prophet (blessings and peace of Allah be upon him) said: “If a person gets sick or travels, the reward for what he used to do when he was at home and healthy will be recorded for him.” Narrated by al-Bukhaari (2834).
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: Thuis refers to one who used to do an act of worship then was prevented from doing it, and his intention was that if it was not for that impediment he would have continued doing it, as is mentioned clearly in a report narrated by Abu Dawood via al-‘Awwaam ibn Hawshab, with this isnaad of the report of Hushaym, at the end of which it says: “… like the best he used to do when he was healthy and at home.” That also appears in the hadeeth of ‘Abd-Allah ibn ‘Amr ibn al-‘Aas, which is attributed to the Prophet (blessings and peace of Allah be upon him): “If a person is doing well in worship then he becomes sick, it is said to the angel who is appointed over him: Record for him (a reward) like that of the action he used to do when he was healthy, until I make him healthy or take him to Me.” Narrated by ‘Abd al-Razzaaq and Ahmad; classed as saheeh by al-Haakim, al-Albaani and the commentators on Musnad Ahmad. Ahmad narrated a hadeeth from Anas which he attributed to the Prophet (blessings and peace of Allah be upon him): “If Allah tests a Muslim with some calamity in his body, Allah says: ‘Record for him his righteous deeds that he used to do.’ If He heals him, then He has washed him and purified him, and if He takes his (soul, in death), He will forgive him and have mercy on him.” [Classed as hasan by al-Albaani and the commentators on Musnad Ahmad]. … According to the hadeeth of ‘Aa’ishah which was narrated by al-Nasaa’i: “There is no Muslim man who prays at night then is overcome by sleep or pain, but the reward of his prayer will be recorded for him and his sleep is a charity for him.” [Classed as saheeh by al-Albaani]. End quote from Fath al-Baari.
If the one who usually prays qiyaam al-layl is overcome by sleep, the reward for his prayer will be recorded for him. So it is more likely -- in sha Allah -- that this applies to the one who wakes up but is not able to pray because of their baby’s crying.
Hence we say to our sister: Be of good cheer and hope for grace and reward with Allah, may He be exalted. Renew your intention every night and resolve to get up, and take measures that will help you to do that. Then if you cannot pray because your baby is crying, you will still have the hope of attaining the reward in full, in sha Allah.
If your husband cannot get up to pray qiyaam al-layl, or he is not interested in doing that, then perhaps he can sleep next to the baby when you get up to pray, so that she (your daughter) will feel that there is someone next to her.
You could also put her in a small crib beside you whilst you are praying, and rock her if she cries, or put your hand on her whilst you are praying, if that will calm her down.
Or you could pray whilst carrying her, if you are able to do that whilst standing, or you can pray sitting and holding her, if you are able to do that.
With keenness and sincerity of intention towards Allah, we hope that Allah will make it easy for you to do what you love of acts of worship and divert distractions away from you.
Part of the etiquette of the believer with his Lord is to think positively of Him. It was narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Allah says: ‘I am as My slave thinks I am and I am with him when he calls upon Me.’” Narrated by al-Bukhaari (7405) and Muslim (2675)
So think positively of your Lord and remember that He will reward you and give to you, and that He will guide your daughter and help you to pray qiyaam al-layl; do not think that He is angry with you or is diverting you from the honour of standing before Him.
We ask Allah to increase you in love for qiyaam and tahajjud and to help you to do that.
And Allah knows best.