One of Warren Buffett's annual reports (I forget which year, but I think it was the late 1990s) had an interesting observation: inflation doesn't affect all firms equally. Instead, it pools where there is a competition bottleneck. If you are the monopoly provider in a market, you have complete and total ability to raise prices in response to your customers having more money available to pay. If you are in a very competitive market, then every time you try to raise prices, some other entrant undercuts you and your customers and suppliers capture the surplus instead.
The Fed, however, measures inflation based on a basket of goods bought by the "average" consumer. Most of these goods are in competitive markets: groceries and gas and consumer electronics. And so the measured inflation rate that the Fed uses to control the money supply significantly undercounts the true inflation rate, with much of the money injected into the economy pooling in differentiated industries like high finance, elite universities, health care, Google & Facebook, etc. From there, it doesn't circulate the way it should, because people in those industries need few goods that the average American produces. Instead, it goes into asset prices, as they try to buy up more future earning potential.
I've suspected that maybe a simple way to fix this would be with "helicopter" Bernanke's crazy idea: drop money out of helicopters. Maybe not literally (imagine the fights on the ground!), but perhaps the Fed could inject money into the economy at the bottom, through direct deposit into consumer's bank accounts or tax refunds, and then collect it from the top, through fees on banks. That way, the money is immediately spent, and so the true effect of the money injected is more easily measurable in the CPI.