I'm not aware of any live stitching plugins for ImageJ -- microscopes usually store discrete images at defined stage coordinates, so stitching plugins will use that as a starting point.
I'm not familiar with the software in the YouTube video. It looks intriguing, but I would recommend asking for a demo to test it on your samples.
Another possibility (I think it's free...) might be Microsoft ICE.
Depending on the software cost and quality, you might do better to send the slides out to a service for scanning. Our Molecular Pathology core facility in NYC does it for less than $6 per slide (scroll down to Leica SCN400) for external users. Some university in your area might be able to do it for even less.
Hope this helps.