Can you simply select the closed requests in the timesheets and it should allow you to charge time against it. The user will need to remember to choose 'Yes' on the 'Include Closed?' radio button option.
So, in the timesheet choose 'Add Requests' > then select the filters if required, and select Yes for Include Closed? option, then search for the requests and it should bring up the closed requests as well.
I don't think there's a restriction or setting to set the length of time after a request is closed to charge time against.