No they're rock solid once they're boxed. And really if the flex is noticeable then something is probably cracked.
I'm not going to say you can't box the steering mount in with the K-frame still in the car, but it's a lot easier with it out. The engine is going to have to be out regardless. As far as a weekend job, probably not. If the car is all together you'll have to pull the engine, drop the suspension and K member, weld it all back up, and then reassemble the whole mess. I suppose you could probably just box the steering mount with it in the car with the engine out, but if I were doing it I'd just pull the whole thing and seam weld the K, gusset the torsion bar pivots and do the steering box mount all at once. It'll be a lot easier with the K sitting on a bench.