I found that there was a kink in the cord and that the metal wire inside of the plastic cord had actually came out. Thankfully I already had a spare cable, which I bought to remove and replace the previously broken handle. The challenge with replacing the cord was getting the old one unscrewed, because the part with the screws was trapped between two metal bars. I had to figure out how to completely remove the arm of the couch in order to access the screws, and the arm was not only attached by three hard-to-reach bolts but also by the couch fabric. So even after separating it I couldn't pull it completely off and had to pick up a right-angle screwdriver at Harbor Frieght to get in and remove the screws.
It was a huge pain but I finally got the new wire in, swapped the handle back to the new wire, and got the arm reattached. The part of the wire which attaches to the release mechanism under the couch was a little long so I had to loop it through one hole to shorten it by like a centimeter. I also stapled the cord to the underside/inside of the couch so it wouldn't get caught in any of the movie parts. Once it was all done I pulled the handle and the recliner released as expected. And as a bonus it's now super easy to pull the cord, taking very little effort. So I guess in the end this turned out to be a good thing.
