Let me try it this way. This is written for macro students, and their teachers. But it's aimed at researchers too. It's about the role of money in macro models.
Ignore what New Keynesian macroeconomists say about their own models. Listen to me instead.
Start with the second-year textbook ISLM model. The price level P is fixed, so real and nominal interest rates R are the same thing. The nominal money supply M is also fixed. Which means the real money supply M/P is fixed too, and the LM curve slopes up. And the IS curve slopes down.
How could we convert that standard ISLM model into a New Keynesian model?
Assume that nobody uses paper currency, and everybody has a chequing account at the central bank instead. That makes it administratively very easy for the central bank to set the interest Rm it pays on those chequing account balances. (Assume the central bank has nationalised the commercial banks, and has abolished currency.) This means the opportunity cost of holding money is now (R-Rm) instead of R like in the textbook model (which implicitly assumes money pays 0% interest). This gives the central bank a second instrument of monetary policy. It can shift the LM curve right/down by increasing M (using Open Market Operations), just like in the standard model. But it can also shift the LM curve right/down by cutting Rm (which makes people less willing to hold money, for any given R and Y, so is like a fall in money demand).
Now bear with me for a minute (I will come back to this assumption below) and assume that the marginal propensity to spend out of income is one, not less than one like your textbook assumes. That makes the IS curve horizontal. Don't believe me? Do the math. Because the slope of the IS curve equals [(1 - marginal propensity to spend)/interest elasticity of spending].
So we now have a horizontal IS curve and upward-sloping LM curve. And you know what that means: the IS curve determines the rate of interest (call it the "natural rate" R*); fiscal policy doesn't work; and monetary policy determines Y at that rate of interest.
"But", I hear you ask, "why should we assume that the marginal propensity to spend is one?". Well, it depends on whether we have got permanent income or current income on the horizontal axis, doesn't it? If it's permanent income, it's quite plausible to assume that everything just scales up in proportion, so if permanent income doubles then consumption and investment spending should both double too, just like if our country annexed a second identical country. But if it's current income on the horizontal axis, and we hold expected future income constant, then sure, the IS curve will slope down just like in the regular model, because the marginal propensity to spend out of current income is less than one. And an increase in expected future income shifts the IS curve to the right, by exactly the same amount.
Now let's drop the assumption that the price level is fixed, and replace it with an expectations-augmented Phillips Curve. And we must now remember to distinguish between real and nominal interest rates. Investment and saving depend on the real interest rate, but the central bank sets a nominal interest rate that it pays on chequing accounts. So we need a vertical wedge, whose height equals expected inflation, stuck between the LM and IS curves, to the right of where those two curves cross, to determine Y. If expected inflation increases, the size of that wedge increases, and Y increases too (holding constant M/P, Rm, and expected future income).
Got it? All you need now is some math. But math is not my comparative advantage, so I will leave that to you.
There is just one difference between my modified ISLM model above and the textbook NK model. My model has got M in it, and Open Market Operations in it. The textbook NK model doesn't. That's a problem for the textbook NK model.
1. The first problem is that the textbook NK model is less realistic than my model. I can talk about Open Market Operations (aka "Quantitative Easing") and the textbook NK model can't.
2. The second problem is deeper. If we don't have M in the model, then how is permanent income determined? It's no good saying that permanent income is determined by the supply side, and must equal the natural level of output Y* in the expectations-augmented Phillips Curve; that's just begging the question of whether there will be enough permanent demand for that level of output. There won't be enough permanent demand for permanent output, if M/P isn't big enough. With a horizontal IS curve for permanent Y, we need an upward-sloping LM curve to pin down a long run equilibrium. My model has an upward-sloping LM curve. The textbook NK model does not. It just assumes, with no justification for that assumption, that permanent income equals Y*, and so current income only diverges temporarily from Y* if the central bank makes random mistakes in setting Rm. Keynes wouldn't like that assumption, because it means just assuming long run full employment, with no explanation of why the economy might get there.