"I would like to transfer the "basis" of the remaining non-deductible contributions into a Roth IRA. "
You can't do that.
Suppose your current IRA balance is $100,000 with $10,000 non-deductible basis (10%). You can't only convert the non-deductible basis to a Roth. If you convert 10% of your IRA to a Roth ($10,000), 10% will be non-taxable ($1,000) and the rest will be taxable ($9,000), leaving your old IRA with a $90,000 value and a $9,000 non-deductible basis, maintaining the 10%.
If you want to convert the entire IRA to a Roth, that will clear the non-deductible basis and the entire remaining portion will be a taxable conversion.
