The first thing I did was remove the plunger.
The purpose of the check valve below the plunger is to allow fluid to flow from the reservoir into the cylinder when you raise the handle, but not allow it to return to the reservoir. Then when you press the handle down, fluid goes into the small hole which leads to the check valve in the bottom of the jack body.
If this check valve didn't check, then the jack wouldn't jack when you pump the pump.
The next think I did was to loosen the top. This required a good amount of torque, so I put a pipe on the end of my largest pair of channel lock pliers while my bench vice on my truck bumper held the base.
Here's the piston that came out after loosening the top of the jack.
Sure enough, the check ball and retainer were loose in the base of the jack. Looking down into the cylinder of the jack, you can see that I put the check valve back in place with the plastic retainer holding it in place.
That piece of plastic at about the 2:00 position goes into a hole that's maybe less than a centimeter deep. If I was thinking faster, I could have tapped the hole for the retainer and used a screw to hold a piece of metal over the check ball instead of re-using that plastic piece. That way I wouldn't have this problem again. I think the plastic piece might come out again.
The hole at the bottom of the above picture goes to the drain valve. It goes to the hole on the left in the picture below to re-fill the reservoir as the piston is lowered. The hole on the right in the reservoir section in the below picture is where the fluid leads to the check valve at the base of the pump.
Re-assembly was easy and straightforward. I filled the reservoir with basic motor oil and checked the operation. It worked like new, no problems. (It's not actually lifting anything in this picture.)
And that's how I saved $40 this evening.