Calculate Turn Around Time

Sure thing. I will use your first table as the starting point.  You can use excel to build a date table and then import, or use the CALENDAR function in DAX to create one.  I went with the DAX.

Calculations-->Modeling-->New Table.  Use the CALENDAR function.  Which needs a start and end date (which will cover in a sec).  Now, could use CALENDARAUTO which will find the start and end data automatically, but does so by searching the entire data model.  Would have worked fine in this small example, but in bigger data models that have date place holders as something like 1/1/9999, CALENDARAUTO will find that as the max.  So back to our CALENDAR function.  Need to find the Min and Max.  We wil do this by search each the Contact Date and Date writting columns and use the Min and Max of those two columns:

Date = 
CALENDAR(
	MIN (
		MIN (TurnAroundTIme[Contact Date]),
		MIN( TurnAroundTIme[Date Written])
	),
	MAX (
		MAX (TurnAroundTIme[Contact Date]),
		MAX( TurnAroundTIme[Date Written])
	)
)

Then add a calculated column for Day Name and then one to label using that day name to denote Weekday or Weekend:

Day = FORMAT('Date'[Date],"DDDD")

Day Type = 
SWITCH(
	'Date'[Day],
	"Saturday", "Weekend",
	"Sunday", "Weekend",
	"Weekday"
)

Here's the end result:

Then need to mark as  Date Table so DAX knows to use that for the built-in time intelligence functions:

Now that we have a calendar that we can use, setting up the DAX formulas should be much easier.   There's actually two ways we can do this, one using FILTER and one using DATESBETWEEN.  

TaT using Filter = 
CALCULATE(
    COUNTROWS( 'Date' ),
        FILTER( ALL('Date'),
		    MAX( TurnAroundTime[Contact Date]) <= 'Date'[Date]
		    && MAX( TurnAroundTime[Date Written]) >= 'Date'[Date]
            && 'Date'[Day Type] ="Weekday"
	    )
)

TaT using DatesBetween =
 CALCULATE(
	COUNTROWS( 'Date' ),
       DATESBETWEEN('Date'[Date],
		MAX( TurnAroundTime[Contact Date]) ,
		MAX( TurnAroundTime[Date Written])
    ),
    'Date'[Day Type] ="Weekday"
)

DATESBETWEEN leverages the use of the time-intelligence, but requires some additional logic to not give a figure when there is no Contact Date or Date Written field:

TaT using DatesBetween = 
IF (
    NOT
        OR(
         ISBLANK(MAX( TurnAroundTime[Contact Date])), 
         ISBLANK(MAX( TurnAroundTime[Date Written])
         )
    ),
    CALCULATE(
	    COUNTROWS( 'Date' ),
          DATESBETWEEN('Date'[Date],
	    	MAX( TurnAroundTime[Contact Date]) ,
	    	MAX( TurnAroundTime[Date Written])
      	),
    'Date'[Day Type] ="Weekday"
    )
 )

Then the final output: