The broken hose and attached NRV do indeed to be part of the blow by circuit. Can't you replace the short broken section, with a short piece of oil resistant hose of similar dimensions?
This does not appear to be the cause of your power loss. Black smoke usually means too much fuel for the available air supply. Perhaps after repairing the broken hose, you may be able to locate an air leak which will probably need the engine running. Is it possible that one of the intercooler hoses is split?
Had you driven far before losing power? Mice nesting in the air filter can cause havoc. Easy to check.
Another possibility may be the EGR valve stuck open. I am not certain as to what the overall effect would be, but it would certainly reduce the oxygen available for combustion. I am quite happy to be corrected.
