C#

• Read the first time value, that is the starting time of work.
• Read the next time, that is the ending time of work.
• The difference between those two is added to the total work time.
• Set the end time as the new start time, that is the starting time of idle.
• Read the next time, that is the ending time of idle (and the next starting time of work).
• The difference between start and end should be added to idle time.
• Set the end time as the new start time (of work).
• Repeat the process until all times have been dealt with.

DECLARE @Data TABLE
(
    TimeLogged    datetime2(0) NOT NULL,
    Action        varchar(5) NOT NULL
);

INSERT INTO @Data
VALUES
    ('20140903 12:50:15', 'start'),
    ('20140903 13:45:25', 'idle'),
    ('20140903 13:56:06', 'Start'),
    ('20140903 16:30:35', 'idle')
;

SELECT
    D1.Action,
    Sum(DateDiff(second, D1.TimeLogged, D2.TimeLogged)) As DurationInSeconds
FROM
    @Data As D1
    CROSS APPLY
    (
        SELECT TOP 1
            TimeLogged
        FROM
            @Data As D2
        WHERE
            D2.Action != D1.Action
        And
            D2.TimeLogged >= D1.TimeLogged
        ORDER BY
            TimeLogged
    ) As D2
GROUP BY
    D1.Action
;

/*
Results:

Action	DurationInSeconds
idle	641
start	12579
*/

SELECT
    D1.NameOfYourActionColumn,
    Sum(DateDiff(second, D1.NameOfYourTimeLoggedColumn, D2.NameOfYourTimeLoggedColumn)) As DurationInSeconds
FROM
    NameOfYourTable As D1
    CROSS APPLY
    (
        SELECT TOP 1
            NameOfYourTimeLoggedColumn
        FROM
            NameOfYourTable As D2
        WHERE
            D2.NameOfYourActionColumn != D1.NameOfYourActionColumn
        And
            D2.NameOfYourTimeLoggedColumn>= D1.NameOfYourTimeLoggedColumn
        ORDER BY
            NameOfYourTimeLoggedColumn
    ) As D2
GROUP BY
    D1.NameOfYourActionColumn
;

Richard Deeming wrote:

But I don't.

Bastard Programmer from Hell
If you can't read my code, try converting it here[^]