What version of SQL Server are you using?  SQL 2012 and above have the functions LEAD & LAG which you could use:
select
    *,
    DATEDIFF(MINUTE, LAG(LogDateTime, 1) OVER (PARTITION BY [ID], [TransactionID] ORDER BY [ID], [TransactionID], LogDateTime), LogDateTime) AS [MinuteCnt]
from [dbo].[TransLog]

From here you can GROUP and SUM however you like.

FYI, the data you provided violates your primary key constraint.