No, the 11th+ Paladin doesn't gain levels in Blackguard, she trades Paladin levels for Blackguard levels. If the now ex-paladin chooses to do this she'd become a 10th level Blackguard/1st level ex-Paladin and have an additional 1 Smite Good/day.
Quoted from SRD:
[11 or more: A fallen paladin of this stature immediately gains a blackguard level for each level of paladin he trades in.
The character level of the character does not change. With the loss of paladin levels, the character no longer gains as many extra abilities for being a fallen paladin.]