What can I do to avoid paying the large federal tax burden?
Unfortunately, there is nothing you can do. Most if not all of this income is falling well above the 10% tax bracket, likely most if it in the 24% tax bracket, so the $52k balance due seems quite believable. This amount of tax underpayment is also likely to result in a tax underpayment penalty and, if your wife is not eligible for an exception to the early-distribution penalty and other funds were not substituted to complete the conversion or rollover of the entire gross amount, also to a 10% early-distribution penalty on the portion withheld for taxes.
If you are unable to afford to pay the balance due, you can potentially arrange with the IRS for a payment plan.
