As the saying goes... Cheap, Easy, Good. You can have 2, pick 2.

Can't really tell how the legs attach to the desk. Is there a way you could get a piece of plywood, actual plywood not particle board or OSB, and sandwich it on the other side of the particle boars screwing through plywood, through particle board, and into the center of the leg? This plus adhesive on all contact points may be sufficient. Obviously you would want to paint any exposed wood.