Mixing ext and DataRegions can be a real pain, since DataRegions are not a true Ext component. It's difficult to coax Ext to size the panels properly. I have three suggestions:

1) if you can get away with an Ext grid, nesting will work more automatically. With Ext grids, especially Ext3 grids, it can be a pain to make them look very good, and they will not automatically inherit all the features of a LabKey QWP. However, just showing the data will be much easier.

2) If you're using Ext3 (which you probably are if you dont know the answer to that), then place a panel inside the window and render the QWP to that panel's body element. Note: the panel will need to have been rendered first. Check out EHR.ext.QueryPanel in the EHR module. The file is under /customModules/ehr/resources/web/ExtContainers.js. This will show you how to defer loading the QWP until after render.

3) if you're using Ext4 (which I recommend), check out LDK.panel.QueryPanel in the LDK module. This is designed to be more of a standalone and it might work out of the box for you. The file is under /unsupportedModules/labModules/ldk/resources/web/QueryPanel.js

If you use this panel directly, it should be pretty automatic, although I've never tried this inside a window. it does get around most Ext layout problems, so far as i can tell.