The pro-rata rule applies to the combined balance of all your traditional IRAs, it is not calculated per account or per brokerage. However, your accounts are not combined with your spouse, since IRAs are "individual" accounts.
To make the "backdoor" Roth work, you would have to convert all your traditional IRA balance to a Roth IRA and pay the tax. Likewise, your wife would (separately from you) convert all her traditional IRA balance to a Roth IRA.
As @dmertz suggests, you might start by converting all your wife's pre-tax IRAs to a Roth IRA. There would seem to be less tax owed if she only has 3 years of contributions. Then, for every year in the future, she can do a "backdoor Roth" tax-free, since her pre-tax balance in traditional IRAs will be zero after the first conversion.
In your case, the only way to do a true backdoor Roth would be to convert all your IRA balances to Roth IRA. You might do that spread out over several years, but you would have to balance the immediate tax impact against the long term growth potential and tax-free future. Alternatively, you can make non-deductible contributions to a traditional IRA and just leave it there until you retire. You will keep track of the non-deductible basis in the account using form 8606 on each year's tax return, and when you start withdrawing from the IRA in retirement, part of your withdrawal will be non-taxable. It's more complicated to keep track of, but it avoids paying a large tax bill for conversion now.
Another idea that surfaced in this forum a couple of days ago was a person who had the ability to rollover their IRA into the 401k plan at their current employer. If you could rollover all your pre-tax IRAs into your current 401k, then your balance of traditional IRAs is zero, and you could then make a true backdoor Roth conversion. However, since any rollover to empty your IRAs won't happen until calendar year 2024 (it can't be made retroactive), you could only do a backdoor Roth in 2024, not for 2023.
You might want to take advice from a professional financial planner who can run some numbers for you.
Incidentally, remember that in 2026, all tax rates are scheduled to increase, unless Congress passes a new tax reform law, so any conversion that you might want to do will probably be cheaper in 2024 and 2025.
