Try replacing 
EXEC sp_executesql @sqlQuery
with
PRINT @sqlQuery
Then you will see what it is attempting to execute.

I think you will half the calls to your function if you use a cross apply:
SELECT k.KENNEL_NO AS KennelNumber,
 ISNULL(k.KENNEL_STAT, '') AS KennelStatus,
 ISNULL(k.kennel_substat, '') AS KennelSubstatus,

Do you have 1 row per Timestamp, TagName? If so I think your query is probably about as fast as it will go. If you have many rows per Timestamp, TagName...

It might be worth trying to rewrite the query. The query below is equivalent to your query and will produce a different execution plan:
It would be good if you...

UPDATE A
   SET A.Action = IIF(B.EmpId IS NULL,'create', 'update')
  FROM dbo.DutyA A
  LEFT JOIN dbo.DutyB
         ON B.EmpId = A.EmpId
      ...

8 hours sounds too much that simple query with that execution plan.
How many rows are on your table?
How big are the columns?
How many rows do...

If you want to union the internal and external downtime together to produce one set of downtime you could use a query like this:
;WITH AllDownTime AS
(