How does the IS-LM model help economists in the day to day analysis of the economy? There are just too many assumptions that are not there in the real world.
The best economic models make useful predictions with as little complexity as possible -- in other words, they make assumptions that aren't "true" in the real world, but make predictions that nevertheless turn out to be true.
For instance, advocates of the day-to-day use of the IS-LM model such as Paul Krugman point to the analysis of liquidity traps as an important case where its users have proved to be right where many other analysts have made bad predictions.
Those of us who understood IS-LM and took it seriously declared in advance — in late 2008 and early 2009 — that big deficits and huge increases in the monetary base would lead neither to soaring interest rates nor to soaring inflation. This was very much not what many influential people were saying: there were widespread forecasts of soaring rates and soaring inflation. Later, there were many people denying that austerity would have negative effects on output.
He provides a more detailed explanation of an IS-LM analysis of liquidity traps (and a few other insights that the model provides).
Two vies on IS-LM, Keynesian and Monetarist, analyses extreme situations of the economy. It provides comparative-static equilibrium condition. It ignores time-lags. It doesn't consider effects of changes in aggregate demand on output and prices.
Keynesian version assumes price level to be constant and inflation ignored. Neoclassical version considers full-employment and so price level is result of nominal money supply and output is exogenous.
As stated in this answer, (that I posted in another question earlier today) Modern Monetary Theory (MMT) considers the IS-LM model to be fatally flawed.