If this is a newly built application, then maybe it's not too late to add an extra field (let's call it "leaveMonths"). This "leaveMonths" field will store text value like following (from your first example):
If this is an existing application, then I'd suggest you just write a simple LS agent to insert the leaveMonths to each existing Timesheet document.
Once the above is done, create a Notes View which the first column is the leaveMonths (make sure it's categorized and sorted).
Then, on your viewpanel, compute the "Filter by category name" with the value for example "viewScope.selectedMonths".
And finally I assume you page should have the following 3 controls:
||to select the first month
||to select the second month
- set the value of viewScope.selectedMonths to something like 082015-092015 (based on what you select using both the DateTimePicker)
- Refresh the whole page
The above is how I would approach the problem if my understanding is right. I didn't do an actual test. Although I'm not quite sure why on one situation you can select just 1 month (July) and on another situation you can select 2 months (August and September). Will there be another situation where you can select 3 months, 4 months, and so on? If the no. of months you can select varies then maybe my approach is not suitable.