Oh, let's address this "conversion/contribution:"

It's also a mess.

"converted $840 IRA contribution ($672 indirect rollover plus $168 out-of-pocket contribution)"

If what you first described is true, the rollover of the net amount + a make up of the withholding would all be Rollover. Or, it is all Contribution. It can't be a split, because it is all from the same action.

"to Roth IRA"

Coming from a Trad IRA, it's never segregated like you just described it. Once it is in the Trad, it might be Basis and pre-taxed, but your taxpayer should have not had basis from the Gross event already described.

"(backdoor Roth"

It's not backdoor. Backdoor is when your taxpayer contributed nondeductible to a Trad IRA and has no other IRA, SIMPLE IRA or SEP IRA funds, and immediately converts to Roth, thereby avoiding any risk that there is taxable income in that "direct transfer to a different tax treatment."

Your taxpayer either had pre-tax or post-tax funds to start with. You already stated you thought the funds are split, and a backdoor can't be split.

"her income is above threshold"

Thresholds only apply for new contributions. What you started with is either Rollover, Conversion, or both. These never have a threshold. There are restrictions for frequency in some cases.

"and IRA contribution are nondeductible)."

Technically, you didn't have any contributions.

"It looks like $168 is nontaxable because it's after-tax money contributed"

No; she made up the missing amount that was withheld. It would be from any money she has on hand, but it was replacing whatever the original account held. It still is considered part of the Gross Distribution.

"and $672 is taxable to convert to Roth IRA?"

See, a Backdoor would not be taxable. And that's what you do not have, in this example.

"(She had a Form 5498 for traditional IRA showing contribution of $168"

It shouldn't be contribution if it is part of rollover. Contribution would also have a "for XXXX" year identified to it. But both rollovers and conversions are simply "funds already in these accounts, being moved or changed as to type of account."

"and a Form 5498 for Roth conversion showing amount $840)"

Yeah, nothing how it's been stated, is right. If that Form 5498 is correct, then:

Your taxpayer still made a bad rollover and everything should be undone.