are you sure your lever is two speed and not just a differential lock? The left side almost looks like it could be… can’t tell for sure, though. Maybe you can post another picture of the the two-speed shifting lever and how it is connected to the axle? Or another picture from the front?
If it was a diff lock, you’d be done on the spot; it would lock the left side axle to the revolving housing, thus locking the spider gears as well and engaging the right side to the left side. Have you used that 2 speed shifting already? If you have, and it worked, forget about this comment.
If not, try this:
If your one side is still bolted to the frame and the chunker is not hooked up to a tractor, you can turn your free chunker side by hand (carefully), can’t you? If you now “shift” your 2-speed-lever, can you still turn it by hand? If not, loosen up the other end and see if you can do it then. If yes, you are done! (By locking you would have converted your differential to a simple bevel gear…)
EDIT: Have been thinking about this all morning at the back of my mind (we have almost noon now here in Germany). The longer I think about it, the more I think the left side is a differential lock. If I was to put a two-speed gearbox in my driveline, I would put it right where the driveshaft meets the rear axle (before it meets the big crown wheel we see in the picture), because there I have the higher rpm of the driveshaft compared to the axles (hence less torque for the same amount of power to be transmitted - I can use smaller gears, so I save space, weight and $$).
Putting some sort of planetary gear on the axle or the differential would at least be heavier, more expensive and more complicated, me thinks. Therefore, I would not do it that way, but again:
If you know for sure the left side is a two-speed gear step, forget this comment and never mind…