My first port of call would probably be to spend a good amount of time looking for a clearer harder area amongst all of the silt, either with a marker or just casting a lead around and feeling for it hitting the bottom.
If there really are no clear areas and the silt is quite soft, to fish running rigs, you'll need to make sure that you use a large running ring (have a look at the Solar ones) as a smaller one could quite easily get clogged up with silt and make your running rig effectively a fixed or semi fixed setup.
Don't use fluro/stiff hooklinks (unless you're fishing a choddy or stiff hinge rig) as they may end up sticking up out of the silt at a really unusual angle (due to the lead sinking into the silt) and the fish will sus you. Either use a supple braid all the way through, or a coated one with a decent amount stripped back, possibly a bit longer than you would normally fish on gravel too.
Feather the lead all the way down through the water to stop it diving deep into the silt on landing too. Use a lighter lead than you may otherwise use as well.
The main thing is to make sure your bait isn't burying itself in the silt and is sitting naturally. You may even want to try some 'natural' baits as the fish will predominantly be feeding on bloodworm and such if the entire lakebed is silty. These would be things such as worm/maggot or even prawns.
Also, the chod or 'silt' rig is slightly different from a standard heli rig (although the lead setup is similar) and was actually designed to fish in this kind of situation. Perhaps you're not quite setting it up correctly? It's never given me anything but very solid hook holds in the botom lip (and is known for it's potent hooking ability). It's used wrongly a lot of the time, but this sounds like an ideal situation for its use.