Yes. Having your period does not close Laylat al-Qadr to you. You cannot pray or fast, but dhikr, du'a and listening to or reciting the Qur'an stay open to you, and the du'a the Prophet (Salla Allahu 'Alayhi Wa Sallam) taught 'A'ishah for this night is one you can make in full. What waits for purity is the prayer itself, touching the Mushaf, and i'tikaf inside the mosque.
What the night is, and why it matters
Allah says: ﴿لَيْلَةُ الْقَدْرِ خَيْرٌ مِّنْ أَلْفِ شَهْرٍ﴾ "The Night of Qadr is better than a thousand months" (Qur'an, Al-Qadr 97:3). The Prophet (Salla Allahu 'Alayhi Wa Sallam) said: "Whoever stands in prayer on Laylat al-Qadr out of faith and seeking reward will have his previous sins forgiven" (narrated by al-Bukhari and Muslim). 'A'ishah (Radi Allahu 'Anha) reported that he said: "Seek Laylat al-Qadr in the odd nights of the last ten nights of Ramadan" (narrated by al-Bukhari). So it falls on one of five nights, and you do not know which in advance.
What your period takes away, and what it leaves
Abu Sa'id al-Khudri (Radi Allahu 'Anhu) narrated that the Prophet (Salla Allahu 'Alayhi Wa Sallam) said: "Is it not the case that when she menstruates she neither prays nor fasts?" (narrated by al-Bukhari and Muslim). So the night prayer is not asked of you while you are bleeding, and it is not made up later; see can you pray during your period.
That is the only part of the night that hayd removes from your list. The rest of worship stays open, as the Prophet's own words to 'A'ishah (Radi Allahu 'Anha) show. When her period came during Hajj he told her: "Do what the pilgrim does, except do not circle the House until you are pure" (narrated by al-Bukhari and Muslim). The restriction is named, and everything outside it is left open. On Eid he also told the women who were menstruating to keep away from the prayer-place but to "witness the good and the du'a of the Muslims" (Umm 'Atiyyah, narrated by al-Bukhari and Muslim).
The du'a for this night
'A'ishah (Radi Allahu 'Anha) asked: "O Messenger of Allah, if I know which night is Laylat al-Qadr, what should I say in it?" He said: قُولِي: اللَّهُمَّ إِنَّكَ عَفُوٌّ تُحِبُّ الْعَفْوَ فَاعْفُ عَنِّي "Say: O Allah, You are Pardoning and You love to pardon, so pardon me" (narrated by al-Tirmidhi 3513 and Ibn Majah 3850; graded sahih by al-Albani). Nothing in it needs wudu or ghusl, and it is the du'a this night is best known for.
What you can do on the night
- Make dhikr and du'a, including the du'a above, and ask for what you need.
- Listen to the Qur'an, or to the night prayer broadcast from a mosque.
- Recite from memory. Here the schools differ: the Hanafi, Shafi'i and Hanbali schools do not allow a menstruating woman to recite, while Imam Malik and Ibn Taymiyyah do, and the Saudi fatwa committee headed by Ibn Baz allowed it; see can you recite Qur'an during your period for the evidence on both sides. Follow the school you normally follow.
- Read the Qur'an without touching the Mushaf, for example from a stand, or spend the time learning. See can you touch the Mushaf during your period for the difference of opinion on touching.
Jordan's Dar al-Ifta says that a menstruating woman who spends the night in dhikr, du'a and sincere devotion has indeed observed Laylat al-Qadr. How much reward Allah gives is His to give, and no one can state it for you. What the sources do show is that the night is open to you.
The last ten nights and i'tikaf
If you intended i'tikaf in the last ten nights, hayd interrupts it, because i'tikaf means staying inside the mosque and a menstruating woman does not stay in the mosque. See what if your period starts during i'tikaf for how the schools treat the days already spent. You can still spend the nights at home in the worship above.
If your period ends in the last ten nights
Once the bleeding has stopped, make ghusl (see how to perform ghusl after your period) and you can pray the night prayer from that point. The odd nights still ahead of you are fully open. For the fasts you missed while bleeding, see what to do if your period starts mid-fast in Ramadan.
If you want to see ahead of time whether your cycle is likely to overlap the last ten nights, a period tracker for Muslim women that predicts your next period will show you roughly where it falls, though predictions can shift by a few days.
This page is a study aid compiled from cited classical sources. It is not a fatwa. For your own situation, ask a scholar you trust.