There's a good chance that your J40 is like our J44, and has flax packing in that rudder tube. Or it has lip seals in the same place - either way you can unbolt and lift the top flange in flat water with the boat not moving, and you'll see what's in there (lip seals or flax packing). If it's packing you can replace it and retighten the top flange to pack it, if it's a lip seal then Curtis has it nailed, you will most likely need to replace, but you might be able to provide additional sealing on a temporary basis with some judiciously applied caulk.
Having just been in and around and all about this area I have a photo or two of the flax and where it goes:
You can see "flax tracks" on the surface above the shelf there. Below the shelf is bearing, above the shelf is seal, in the case of our J44 it was flax, and we're of essentially the same vintage as you.
In your video I can see the top flange moving, so it could be as simple as tightening the bolts, also, to re-secure the flax. With rudder packing glands there is no need to worry about the flax overheating, so as long as you don't get too much friction, you can tighten that flange down quite firmly.
Good luck, it's no fun seeing the ocean get to the wrong side of the boat, is it!