When unemployment is present, lower wages for some workers can stimulate renewed employment and -- depending on elasticities -- possibly greater purchasing power as well or at least not proportionally diminished purchasing power. (Each worker earns less but there are more workers employed.) There won't in general be much of a deflation. The hiring of some workers can also lead to an upward spiral in production, employment, and again purchasing power, as outlined by W.H. Hutt in his books on Keynes.
Krugman and others wish to argue that the New Deal years were ones of recovery; that is fine but it increases the chance that the Hutt scenario and not the Keynes scenario would apply at that time.
The simplest version of the Keynesian argument on money wages also relies on labor as the primary source of marginal cost (true in many but not all sectors) and lack of market power for retail prices, among other assumptions about market structure. Yet another scenario is that some nominal wages fall and entrepreneurs (with some market power) invest more in response and hold retail prices relatively steady.I believe Keynes's "falling nominal wages-falling prices-constant real wages-constant unemployment" scenario does hold for some of the 1929-1932 period and indeed I have argued as such in print. But once we get into the Roosevelt era, we have government propping up some wages above market-clearing levels and thus higher than necessary unemployment. Note that the Roosevelt policies applied only to some workers and by no means to all or even most workers, which again suggests the Hutt analysis is more relevant than the Krugman/Keynes analysis.
Roosevelt's lunatic cartelization schemes could have been better executed than they were, but even perfectly executed, they would have failed, because it is not possible for the government to prop up all wages and prices; even Soviet Russia didn't manage it. If the nominal price of labor is too high, much of it will move into the black market. It was even harder for Roosevelt, because he lived in an era where record keeping was somewhat slapdash--I was just reading a travel memoir from the fifties where the author goes to apply for a passport and finds he has no birth certificate. Now, if they really wanted to, the government could send someone to the address last associated with your social security number and inquire what you were doing with your days.
Which is a disturbingly good argument agains the most prominent relic of the New Deal. But I digress.