Let's try to make a rough estimate.
Intel's article Intel SHA Extensions gives some details on these instructions as well as sample code. The main feature is the
sha256rnds2 instruction, which performs two rounds of SHA256, out of the 64 rounds that are needed to hash one 64-byte block. A Bitcoin header is 80 bytes long, so that's 2 blocks, and because the mining algorithm is SHA256D, we need to do it twice. So we need to execute
sha256rnds2 128 times to perform one Bitcoin hash.
I'm no expert on modern CPU architectures, but I would suspect a complex instruction like this would take more than one clock cycle; nonetheless, let's generously suppose it doesn't. Since each round depends on the outputs of the previous, these rounds probably have to be executed serially (on each core), so not much parallelization can be done. But let's very generously assume there are resources that can be shared, so that the CPU can execute two
sha256rnds2 instructions per clock cycle. Let's also generously assume that all the ancillary code needed to set up for
sha256rnds2 can be pipelined and doesn't require any additional clock cycles. So it takes 64 clock cycles to perform one Bitcoin hash.
Now, how fast can we run the clock, and more importantly, how much power would be used? Since power consumption is paramount, let's suppose that we would want to use a mobile CPU. The Wikipedia article on Skylake suggests that the SKL-Y-1 model will have 2 cores and a thermal design power (TDP) of 4 W. Let's assume the TDP represents the actual power consumption for our mining application, and furthermore let's neglect the power consumption of all other components of our machine (memory, power supply, etc). There isn't any information given on clock speeds, but Intel's current Core M processors have a base clock speed of up to 1.2 GHz, with "turbo" up to 2.9 GHz. Let's suppose this new Skylake device could run at 3 GHz sustained.
So our overall hash rate is
3x10^9 clocks/sec / 64 clocks/hash * 2 cores = 93.75 MHash/sec
With a power consumption of 4 W, this gives an efficiency of 23.4 MHash/J.
By comparison, according to bitcoin.it's Mining hardware comparison, current ASIC miners are able to produce 1000-2000 MHash/J.
Even under extremely optimistic assumptions about the mining performance of the Skylake processors, modern ASIC devices are still 40-80 times more efficient.