Welcome to the forum!
Boy, have I got good news for you! I had to do this, like ten years ago on my T1560 which has the exact same setup, judging from your video; which by the way was perfect! Your steering/connecting rod has a ball joint on both ends. On the other end of your ball link, the one attached to your steering linkage you should find that the rod sits on top of the ball and therefore can never fall off. Take both balls out and swap ends of the entire connecting rod assembly including the balls so the one that has fallen off will now sit on top of the ball and support the steering rod. But, and here is the important point, you have to grease both of the "maintenance free" balls. Do this by injecting grease under the rubber seal and onto the ball. ESPECIALLY the one that dangles. I use a syringe type dispenser with a very fat "needle" about a 0.7mm hole to push up in there and squirt grease in, wriggle, squirt, wriggle, squirt,,,. I have done this every ten hours or so, actually that's a joke since there is no clock on mower, but I just do it twice, three times a year and the now upsidedown ball socket has never failed.
I hope this makes sense to you. Technical writing, when the reader has no clue what you are (I am) talking about is tough. If you need further assistance someone else may have a better suggestion.
Some people will suggest peening, pinging, whatever the wallowed out socket so the ball won't fall back out. Some will recommend a new rod assembly, which should not cost more than a few hundred dollars. I like my method.
Oh, one caveat if you turn into a ditch, bank, curb or other, the known, bad ball/socket can still pop off. Just stick it back on, remember it is now balls,, err, I mean ball up, and mow. I push/pull on my steering connecting rod at the upsidedown ball joint from time to time and it shows no sign of wear, much less of falling off.
