Why was it so difficult to find an effective solution to inflation problems between 1964 and 1983?

It is due to the nature of economic policy. Normally inflation and unemployment are inversely related, so policy decisions can be made to cure one at the expense of the other (for instance, raising of interest rates lowers inflation but risks stifling business growth). During the period between 1964 and 1983, we experienced "stagflation" (high unemployment AND high inflation). So when we experience both at the same time, policy makers have their hands tied as to what to do. If they decide to try to get inflation lower, they risk making unemployment worse (and it's already bad) and if they try to get employment lower, they risk making inflation worse.