Doing a Google search of the problem, I found this Instructable: http://www.instructables.com/id/Fix-a-dyson-vacuum-hose/#step1
It worked like a charm - at least for now!
Since I was successful with that, I decided to change both the filters. The pre-motor filter was easy, as this is meant to be cleaned every six months anyway. The HEPA filter was a pain, but after some careful prying with a small screwdriver, I managed to get that done without breaking anything.
Another broken part on the vacuum was the clip holding the pre-motor filter in place. I ordered the new clip...... and couldn't get the broken bit out. Mike helpfully stepped up to the plate and muscled it out, again without breaking anything.
So now, the Dyson is working like new! Good thing, since the new pup is already generating hair tumbleweeds.