I . . . bought some 5800 105's. Going to see what levers I can get to fit them
My recommendation is to fit the 5800 calipers to the rear, leaving the cable and brake lever as is; and go and try it out. If the power and modulation seems OK then do the same on the front. And trial again. I have measured the cable pull on levers of a similar vintage to your 600s and they pull 9mm (but could pull more). This effects movement in the single pivot calipers of about the same: the blocks are further from the rims than they should be. This length of pull is more than enough to achieve effective braking using 5800 calipers. The match up may not be optimal, but provided the modulation is acceptable to you and the lever/caliper achieves braking effect of sufficient strength for your hand size and finger strength then I would not spend significant money on more levers, and certainly not STIs (just to use as brake levers).
Edit: Off topic but you may find that replacing the current blocks/pads with a set of quality ones with your current single pivot calipers achieves the improvement in braking that I assume is your ultimate aim.