Did you check the report in LSI by giving the specific date range?
Content -> All Categories -> Members (tab under 'Categories Overview').
This report gives the list of members (signed-in users) who have visited the community and performed some activity. The number of visits against each user may serve your purpose. The report shows only Top 50 users but you can get the list of all users when you export the csv file.
Alternately, this can be achieved if you have the setup to store bulk data csv generated by Lithium.
But I still wonder why the numbers in the LSI -> Content -> Members -> Export CSV and in the Admin -> Metrics -> User Reports -> Registered User Sessions differ so much...Which I should use?
I found the data in LSI to be too limiting as it caps out at 2000 results. We have far more active members than that so that report is not suitable, as a result we use metrics in community admin to get this figure. It would be nice if there was not an arbitrary limit to the results in LSI.