Why should the Fed remain independent from political authority?
The reason that the Federal Reserve System (the Fed) should remain independent of political authority is that it needs to be independent so that it can make its decisions based on what is best for the economy in the long run, not what is politically expedient.
A good way to think about this is to look at the current fiscal situation that the elected branches of government have created. It is to some clear that something has to be done to reduce our deficits. We seem to need to reform entitlements such as Medicare and Social Security. This seems necessary for the long term health of our economy. Nothing has been done to address this. The main reason for this is that it is not in the short term interests of either political party to advance such reforms. Any party that does so risks losing elections as the other party claims that no reforms are necessary (which is a position voters typically like).
What this shows is that elected branches of government are not experts at long-term economic analysis. They are motivated by the need to win the next election. Therefore, if the Fed were under political control, it would be forced to take actions that would not be helpful to the economy. It would be forced to do what was needed to win elections for the party in power, not what was best for the economy.