All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Firstly, this woman's Ihraam (ritual state of consecration) was valid and there is no problem with it because the purity from menstruation or Nifaas (Post-partum bleeding) is not among the conditions for the validity of the Ihraam for Hajj and ʻUmrah.
Asmaa’ bint ʻUmays gave birth to her son Muhammad ibn Abu Bakr during the Farewell Hajj. The Prophet, sallallaahu ʻalayhi wa sallam, commanded her to perform Ghusl, wrap her private parts in a clothing (wear something over her private parts to prevent the blood from spreading onto her body and clothes) and assume Ihraam.
The Hadeeth in reference was narrated by Jaabir about the description of the Farewell Hajj and it was cited in Saheeh Muslim. It should be noted that Tawaaf Al-Qudoom (circumambulating the Kaʻbah upon the arrival in Makkah) is an act of Sunnah according to the opinion of the majority of scholars and neglecting it does not require offering an animal sacrifice as expiation.
Al-Mawsooʻah Al-Fiqhiyyah Al-Kuwaytiyyah (Kuwaiti Encyclopedia of Islamic Jurisprudence) reads, “Tawaaf Al-Qudoom is an act of Sunnah for the Afaqis (outsiders) who come to Makkah according to the Hanafis, Shaafiʻis, and Hanbalis as a greeting for the Sacred Mosque. It is recommended to perform the Tawaaf upon the arrival in Makkah with no delay. The Shaafiʻis held that it is recommended for all the outsiders who arrive in Makkah to perform Tawaaf Al-Qudoom upon the arrival whether or not they are in a state of Ihraam …”
She is not liable for any animal sacrifice because she performed this Tawaaf while being ritually impure, especially that she did not know of that and believed herself to be ritually pure then.
Allah knows best.