Before staking, it's a good idea to clamp the plate to the housing with a bit of pressure, reasonable pressure, you don't want to crush or break anything.
Notice I'm clamping to the housing, not to the worm gear shaft.
Next I took the assembly over to a vise. I grabbed an old axle stub end for use as a rivet buck. You can use the corner of the vice or what ever you can make work.
I'm resting the opposite side of my rivet face against the "buck"
Now it's just a matter of recreating that flared/expanded rivet shape. I'm going to tighten both rivets while I'm at it. I actually started with the pick end of one of my body hammers, because it happened to be within arms reach and it had the right shape.
I would suggest starting with a blunt tipped centering punch to get the initial flare shape started. Then you can finish off the job with a small ball peen hammer.
There ya go.
I'm tellin ya, that thing is silky smooth now, it's going to work like new.