Hi, we’re doing something similar, not exactly the same though, by using filters. Basically edit your Kanban View and apply a filter, something like that:
done = false || (done = true && doneAt > 'now-14d')
This will show all tasks that are not done or have been marked done in the last 14 days, after which they automatically go away. There’s an additional list view which shows done tasks for us, but that’s obviously optional. Maybe that helps?