"She is going to sleep."
Translation:वह सोने जा रही है।
Only the principal verb needs to be conjugated with the subject. सोना is being used as the indirect object of the verb जाना.
It's in the oblique case. It's similar to how in 'वह मेरे घर जा रही है।' we use 'मेरे घर' instead of 'मेरा घर'. The reason is that जाना is an intransitive verb and there is an implicit postposition ('ghostposition') before it and 'सोने' / 'मेरे घर' is the object of that implied postposition.
Verb infinitives can take the oblique case because they behave as nouns (similar to English).