Yse a DAX measure to calculate the sum of the duration for any events that fall on a given date. You can do this using a disconnected date table. 
For example, if you have a disconnected date table called Date, and your event table is called Event, you can have a measure such as:
Filtered Duration =
CALCULATE (
    SUMX (
        Event,
        DATEDIFF (
            MAX ( MIN ( 'Date'[Date] ), Event[StartDateTime] ),
            MIN ( MAX ( 'Date'[Date] ), Event[EndDateTime] ),
            SECOND
        )
    ),
    FILTER (
        'Event',
        'Event'[StartDateTime] <= MAX ( 'Date'[Date] )
            && 'Event'[EndDateTime] >= MIN ( 'Date'[Date] )
    )
)