Recharacterize a 401(k) withdrawal and separate Roth IRA contribution as a rollover to Traditional IRA?

I am doing my mother's 2017 taxes. She made a mistake last year with a taxable 401(k) withdrawal, and I think I found a way to correct it, per IRS rules. Please let me know if what I am suggesting below will work (or not, or if there is a better way).

What my mother did:

- She cashed out a small 401(k) in 2017 (for $6019).

- She then separately made a Roth IRA cash contribution for $6500 (within 60 days).

What I want to do:

- First, designate the Roth IRA contribution as a 60-day rollover (and taxable conversion) from the 401(k).

- Second, recharacterize a portion of the rollover as going into a Traditional IRA, instead of a Roth IRA. And avoid the taxes on that portion.

My mother is 74, so she can't directly contribute to a Traditional IRA. But she can do a rollover from a 401(k) to a Trad IRA. She separated from that 401(k) employer in 2017, and was not required to take a RMD until separation. She has until April 17, 2018 to take her first RMD from this 401(k), and I'll make the 2017 RMD amount is covered before the rollover portion.