If you're watching a float you can't give due concentration to a ledger rig at the same time.
The old boy who gave me some help when I was starting out fishing said, and I think he's right, that the chap fishing with one rod is fishing with it; the chap fishing with two rods is fishing with neither.....
Mind you, he'd never seen a hair rig, a self hooking rig or a bite alarm and has probably passed away long since in blissful ignorance of these abominations.
I generally do the same as you, two rods, but I always rig the ledger as a self hooker.....
Seems to work. I may miss the odd fish on the ledger that just mouths the bait - I don't see it and don't therefore strike - but so what; the second rod is mainly a bonus fish thing for me. Or, if nothing on the float but lots / better quality on the ledger, an indication that it would be a good idea to try two ledgers and put away the float gear.
Mind, if fishing water where the fish are big enough to pull a rod in then use a reel with freespool / baitrunner facility.
In passing, update from post a year and a half or more ago, I am still trying circle hooks on the ledger, off and on. They don't work with the bait on the hook, that's a cert. Bloody difficult to hook a bait with one in the first place and the narrowness of the gap between point and shank is easily blocked if the bait should move at all. Hair rigged best and only.
They don't seem to be any better or any worse than a conventional hook so far, but then again I must confess not to have used them very much. More a confidence thing, when catching on conventional hooks I stick with them. When not catching on conventional then I try on a circle hook - so hardly a fair test for them if they are only being fished in hard conditions. I shall have to make an effort to fish two side by side, conventional v. circle and then see.