What happened to Kushina's soul
Nothing special happened to Kushina's soul, she died from having Kurama extracted from her body and the wounds inflicted during the battle with Kurama.

Minato used the Death Reaper on himself to seal away the half of Kurama that he had stored in himself. The other half got sealed in Naruto and Minato infused the Eight Trigrams Sealing jutsu with his own and Kushina's chakra.

Here you can find the scene where Kushina tells Naruto that Minato had infused her chakra into the seal: https://youtu.be/D3tJoGr6pp4?t=68