Is Fasting without Ghusl after Period Valid?
Question

My menses ended before Fajr prayer and I waited until after the prayer, but I fell asleep and I woke up at ten a.m. I did not do ghusl because I forgot, and I took my sick daughter to the hospital, and I was busy until the time of ‘Asr prayer, then I did ghusl and made up the prayers that I had missed.
Allah knows that this is the first time this had happened to me since I reached puberty. The reason was that my period usually lasts for 9 days but this month it was 8 days, but I waited to make sure, but I was fasting. What should I do? Should I make up the fast?
Praise be to Allah.If you were certain that your menses had ended and you formed the intention to fast before dawn, even if it was one minute before, then your fast is valid, even if you delayed doing ghusl . 
But if you were uncertain as to whether your period had ended, but you still intended to fast, then the fast was not valid because of your uncertainty as to whether your menses had ended.  
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) was asked about a woman who fasted when she was unsure whether her period was over, then when morning came she saw that it had indeed ended. Does her fast count even though she was not certain that her period had ended? 
He replied: 
“Her fast does not count, and she has to make up that day , because the basic principle is that the menses was still there, and her starting to fast when she was not certain that her menses was over means that she started to do an act of worship when she was uncertain as to whether one of the conditions of it being valid was fulfilled or not, and this means that it does not count.” (Majmu’ Fatawa al-Shaykh Ibn ‘Uthaymin, 19/107)                                                                                       
If a woman knows that her menses has ended , she must do ghusl so that she may pray, and it is not permissible for her to delay that until the time for prayer ends. If she does that, then she has to repent, as well as make up the prayers that she has missed. 
If she forgets that her period has ended – as you mention in your question – then there is no sin on her in sha Allah for delaying her prayer, but when she remembers she must do ghusl and offer the prayers that she has missed – as you did. 
We ask Allah to forgive us and you. 
And Allah knows best.