da, Sql Server - rextester

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--create table
CREATE TABLE DATA2
(
EMPLOYEE_ID INT,
APPLY_DATE DATE,
ITEM_TYPE nvarchar(50)
)
-- insert data
INSERT INTO DATA2(EMPLOYEE_ID,APPLY_DATE,ITEM_TYPE) VALUES
(1,'2017-05-23','Sickness'),
(1,'2017-05-24','Sickness'),
(1,'2017-05-25','Worked'),
(1,'2017-05-26','Vacation'),
(1,'2017-05-29','Vacation'),
(1,'2017-05-30','Worked'),
(1,'2017-05-31','Worked'),
(1,'2017-06-01','Sickness'),
(2,'2017-05-25','Sickness'),
(2,'2017-05-26','Sickness'),
(2,'2017-05-29','Sickness'),
(2,'2017-05-30','Sickness');

with tbl as
(select d.*
	,case when (item_type = lag(item_type) over (partition by employee_id order by apply_date))
			then 0 
			else 1 
	end grp_tmp
from DATA2 d
where 
	item_type <> 'Worked'
)
,tbl2 as 
(select t.*
	,sum(grp_tmp) over (order by employee_id,apply_date 
						rows between unbounded preceding and current row
						) 
	as grp
from tbl t
)
select 
EMPLOYEE_ID
	,ITEM_TYPE 
	,(CONVERT(VARCHAR(24),min(apply_date),103)
	+' - '
	+CONVERT(VARCHAR(24),max(apply_date),103)
	) as range
from tbl2
group by EMPLOYEE_ID, 
		ITEM_TYPE
		,grp
order by 
	employee_id
	,min(apply_date);

View schema
Execution time: 0,02 sec, rows selected: 4, rows affected: 12, absolute service time: 0,16 sec