How to find where the dryer vent is broken or clogged

Ah, how I hate people who build things designed to last forever, or rather to need the house ripped apart when they don't.

I'd try a loop of wire (galvanized steel or aluminum, whichever the duct is, so they don't corrode each other over time) wrapped around the pipe in the hole to allow you to pull on that while shoving the stub pipe into the elbow. Look down through the stub so you can see what you are doing. Attaching that wire to the stub would then hopefully keep the elbow from falling off again. You would need to seal the joint with duct seal since there's no way you can tape in there.

Alternatively, figure out where the access panel for this job should have been put, buy an access panel, cut a hole in the ceiling (below the pipe), and put the access panel there. Or do what too many folks do, and simply cut a hole in the drywall, convince yourself this will never happen again, and patch the hole in the drywall.

If you are having trouble mating the male and female bits, bend the male bits inward a bit more than stock.

Since I my last edit, which has been a few years, I have replaced various components in the dryer, including thermostats, wires, etc. I discovered there were burn marks on the wire terminals and the component it lead to. The failure was fairly consistent and wondered why it keeps happening. Lo and behold, the terminals I was using were insulated crimp terminals and it was increasing the resistance of the wire causing it to burn out. I never suspected it because there was no apparent melting of the plastic insulator. I wanted to use non-insulated terminals, but they were very difficult to find and removing the insulation from the terminal connector made an unreliable connection.