I don't think I have much to add. To avoid any of the $240,000 distribution being taxable, $240,000 would have to have been put into the IRA as a rollover. Since the amounts withheld for taxes was not rolled over, the sum of amounts withheld remains a taxable distribution. You get credit on your federal and state tax return for the amounts withheld for federal and state taxes and you'll get back excess withholding as refunds on your federal and state tax return. However, it's far to late to roll any of the tax refunds into an IRA. The amount not rolled over remains a distribution on which you will have paid the taxes. This is the same as if you had simply taken an amount equal to the withholdings out of your 401(k), put the money in a non-retirement account and paid the taxes on that distribution. It just so happens that the non-retirement accounts where the money was held for a time was your federal and state tax accounts.
As VolvoGirl said, to complete the rollover of the entire $240,000 by the 60-day deadline you would have had to have come up with money from another source while the IRS and the state tax authority had some of your money until refunded.
