QA Sprint Workflow and Jira JQL Filter Configuration

[Donata]

Hello,

I hope my question will be clear.

Situation: we work on 2 weeks' length sprints. QA team test one sprint behind. Meaning, when the issue is moved to 'Done' and Sprint is Closed, they start testing those issues. 
I need to close the sprint and move issues to Done, as otherwise will not have a burn-down chart and see what's been done.

Problem: QA is working (testing issues which were completed last sprint) on a Closed sprint, and I cannot move those closed issues into a current sprint. Neither I can (at least I think I can't), create a separate Sprint for QA, where they will test those 'Done' and Closed items from the previous sprint. 
What I do now is - create a separate filter for them, but there they see the only status in Done. Because I cannot set up a filter with other statuses of those issues, as they are already completed in the last sprint and my filter shows only the current sprint. I am a bit lost, yet I bet there is a smarter way to fix it.  

Expected result: How to set up Software project board, filter and/or sprint so that QA would have a smooth testing process, when testing one sprint behind and would not need to get lost. 

Thank you,

[Deleted user]

I try to explain with the above Pic

1) Development - can be done in Scrum Board

2) QA Testing - can be done in Kanban board