edit: my HERMS coil is movable and not mounted into any vessel.
I do this, but I don't use just ice-water.
By the time my boil is done, the MLT and HLT are empty. I put about 5 gallons of tap water in the HLT and put oxiclean in it.
I put about 3 gallons of tap water in the MLT and put 10# or so of ice in it.
When I am ready to chill, I put the coil into the oxiclean water in the HLT and start pumping. This takes the first 100* or so off wort and heats up that oxiclean water at the same time.
Then I move the coil to the ice-water in the MLT and finish the chilling.
When the beer is in the carboy, I now have 5 gallons of hot cleaning solution and 5 gallons of clear rinse water sitting there ready to go.