If the deposits are really thick and hard - might have to sand-blast it to clean it up. Soaking in solvent or plain water + detergent, power washing generally only gets the soft surface deposits off. I've seen people use ultrasound cleaning units as well - though be tough to find one big enough to do the whole manifold. Some solvents can affect the metal itself, they work via ion exchange can actually weaken the metal. Though most of those types of solvents will also do a number of biologics - so the likelihood of getting your hands on something like that is pretty remote.
Soaking is going to be the simplest option - could try running wire brush over the tough spots first - help loosen it up and allow the liquid to penetrate it. See if that gets off enough of the deposits to look satisfactory. Could be an iterative process - scrub, soak, scrub some more, soak some more, etc. But can't argue the cost of it (almost zero - just your time).
Sandblasting - definitely the fastest way, most efficient way - depending on the media used in the blasting process. Most can be rented for fairly cheap - you just have to provide the blasting media. Stuff like bicarbonates, glass beads, plastic beads, and walnut shells will eat deposits and rust, but not really aggressive enough to eat into the metal. Something like aluminum oxide - have to be careful as it will eat a hole in the metal eventually, but will make short work of the deposits. Even a single pass is enough. I use this all the time with older machine equipment - run a degreaser first, scrape all the loose bits off first, get down to the hard deposits - then blast it. The deposits literally disappear under the media stream. Some media, like the bicarbonate ones, actually "coat" the cleaned metal, prevents it from corroding right away. Can be left on for months, then washed off with plain water to be finished as desired.