Yes, if your wife made a deductible traditional IRA contribution for 2022 then part of her conversion will be taxable. When you have both pre-tax and after-tax contribution then the pro-rata rule applies. This means that with each distribution/ conversion you will have a taxable and nontaxable part.
For the Backdoor Roth to work as indented you must only make nondeductible traditional IRA contributions and do not have any pre-tax funds in the traditional/SEP/SIMPLE IRAs.
You are not getting taxed again. Pre-tax means the money hasn't been taxed. The $6,000 were deductible in 2022 (pre-tax) therefore when you convert it to an after-tax account you will have to pay taxes.
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