Hi @Massive-pace-3774
The child is entitled to a residence permit based on paragraph 33 of the immigration law (Aufenthaltsgesetz), as the child of a person with a residence permit in Germany. Your wife is not automatically entitled to a permit but may apply for one as the parent of a child. This is at the discretion of the authorities, however, and is only granted in cases to avoid exceptional hardship. It can only be granted if she is financially secure (i.e. has enough money to live on).
Your wife can apply for a residence permit as your wife (permit § 29). This is more straightforward but she will have to show A1 level of German and that she is financially secure.
Residence permit § 25,5 is only possible after a person has had the temporary paper called a Duldung for 18 months. She would first have to apply for a Duldung. If she wishes to do so, she applies for a Duldung with the local Ausländerbehörde.
With the permit § 25,5 a person can be self-employed, freelance or set up a business.
If she is not entitled to a permit as your spouse and does not fulfil the requirements for a different type of permit, then applying for a Duldung is the only option I can see left for her.
Best,
Éanna