Every time you write down an IS-LM model you should hear a clock start ticking in your head. The longer the clock ticks, you more you need to worry about this problem because the more that a) the price level may change, or b) expectations about future price level changes will start to matter.
In defense of the IS-LM model, Arlo would like to make the following points.
1. The IS-LM model is a pedagogical device for teaching elementary/intermediate macro. It presents the Keynesian (or Hicksian) view of the joint determination of interest rates and output in a clear way.
2. Assume that the price level is fixed and the term premium is constant, if that will make you feel better.
3. Tyler's point three is about facets of monetary theory that are problematic for reasons have nothing to do with IS-LM.
4. Assume that expectations are fixed, if that will make you feel better.
5. Try explaining Keynes' arguments for why monetary policy is ineffective. Your IS-LM pedagogue will be showing the interest inelasticity of investment as a vertical IS curve and the liquidity trap as a horizontal LM curve at the vertical axis. What have you got?