How much computing power is required for a GPS Device/Station That Uses L1 and L2C Frequency to Calculate For Ionospheric Delay-Interference?
How much computing power ... ?


I am not able to quantify exactly in terms of FLOPS, but it is very cheap. 

An example algorithm for a dual frequency user is described in chapter 20.3.3.3.3.3 of the User Interface specification IS-GPS-200.
The usage of ionospheric data from subframe 4 page 18 for a single frequency user is described in chapter 20.3.3.5.2.5.
I would guess that the computations for the ionospheric model are slightly more expensive than the dual frequency approach.


  At what rate would the GPS need to receive ... ?


The position update rate is irrelevant. A receiver can measure ionospheric refraction without ever computing a position solution.

Dual frequency is not a panacea to enhance position accuracy. In fact, it does not help much until you


use a multipath supressing antenna
use a wideband RF frontend and a high IF sample rate
take coherent measurements on two frequencies (cheap chipsets only allow to switch back and forth)


There is no mass market for these solutions, so they remain expensive. Smartphones need chips that perform in urban canyons and indoors, not centimeter accuracy.