It's a sturdy, yet flexible metal rod meant for running cables. Figure out which of the two openings you can easily pull from, hard. Insert the tape from that end until it emerges on the other side. Then cut holes in a towel and secure it to the fish tape with zip ties or hose clamps, and pull it back through.
The rag or towel should force the brush out of its way on the way back.
In the future, NEVER use brush rods that can unscrew to clean a dryer vent. Only use the kind that are permanently attached to a tough braided wire.
Thank you @Peter Lin for the great idea about using an auger, homemade out of a wire coat hanger, to fish a stuck lint brush out of the dryer vent duct. After many hours and failed attempts using other unsuccessful methods, the coat hanger auger worked on the first try. See the before and after pics. As suggested, I formed the auger by wrapping coat hanger wire around a broomstick and then opened up the top portion to flare out like a cone. I then attached the auger with a metal hose clamp to a long piece of flexible PVC pipe. I inserted the auger end of the PVC pipe up the duct until I could feel it stop on the stuck brush. I then turned the pipe a few times until I felt (prayed) that the auger had secured the stuck lint brush. I then slowly pulled the pipe out of the duct to reveal, hallelujah, the lint brush successfully captured by the auger
Does this vent go vertically into the slab and then make a right angle bend on a radius? If you have a fish tape it would be extremely strong and well suited for pulling this out, but if you just want to use something you have on hand, then you could use a garden hose (not connected; no water, just the hose). You would run it down in the vent to probe and determine how far in the blockage is.
Then rig a hook with coat hanger wire to snag the brush. The hook would be a length of coat hanger wire wound around the hose and secured with duct tape. Of course, anything like this could go wrong and you'd have something additional jammed in.
Another possibility is to use a shop vac on the inside end and run a dry hose in from the outside to push on the brush.
Run the rod thru the opposite way than you were when it got stuck.
Make sure it goes out both ends.
Attach the second brush (or a new one), and make sure it is on tight! For extra security, drill a hole thru the brush and run a long piece of twine or rope thru it so you can retrieve it if gets stuck too.
Pull the new rig out the same direction as you were originally working. Don't twist it. Just pull. Voila!!
In my experience the dryer vent is best cleaned with a combination of compressed air and vacuum, both in industrial calibers. The compressed air breaks up any lint balls and the vacuum sucks the looser lint out. Don't forget to vacuum out the dryer itself.