why is it wobbly?

are the legs firmly attached to the desk? are the points where they screw in, intact. because if this in not the case, whatever brackets you put on to firm it up, is still a temporary fix.

if no start there - screws can be reset ect, youtube superglue hacks to find out how to strengthen screw holes

https://www.youtube.com/shorts/zhEmMMLsR0Y

so if the legs are firmly attached i would use 1/4 " all-thread (threaded rod) and make a diagonal cross brace from leg to leg

drill holes thru the legs using the hex nuts to hold each side , that will triangulate them. you could put the bars low to the ground or higher up.

you are going to need nuts on both sides of the legs, so 8 nuts.

if you want to get crafty with it you could use some sort of tube to sleeve over the all thread. this would help by creating opposing forces.

shopping list.

2 pcs of 1/4-20 threaded rod long enough for the diagonal 8 pcs 1/4-20 nuts. maybe acorn nuts on the outside of the leg, hex nuts for the inside

optional sleeve, could be pvc or emt or copper tube (hard not soft) big enough diameter to hide the nut inside

that's my fix.