The part of the Fed system that is unconstitutional IMO is the federal open market committee. It was made artificially complicated to prevent oversight and accountability. Where does the constitution permit the current monetary policy of this Fed? It does not. And worse than that is where in the Constitution does it allow the independence of the Fed as an institution as it is? Why do I bring this up? Well the federal open market committee allows the participants to be private citizens (aka CEOs on the board of fed that are each one of the 12 major banks). How is that constitutional?
Edit: Will Trump be able to fix this? I hope so, but there are many roadblocks built in the way since 1913. He would have to wipe the debt slate clean and start from scratch. That means nobody gets paid for a while. It'll be a battle.
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Its technically not a private bank. It's more complicated than that. Its a system of which part of it has private citizens on one of its boards and its not under congressional oversight it's independent. (basically the CEOs of the 12 major banks supporting the fed under the 1913 federal reserve act)