How can I quick fix cement steps that are disintegrating?

There is a product that is a cement patch. It works great and is fairly easy to apply. I would give that a try and consider trying something else on the steps or perhaps coating them with a sealer before you use a de-icing treatment this winter.

This video is for a product, sika, I just saw demonstrated last week. I can't vouch for the quality but it seems much easier to use than concrete. They have several products so you might want to check them out.

https://www.youtube.com/watch?v=VaZediAD8YI

The other option would be to buy bag(s) of readymix that is a just add water product. Pour into a wheel barrow and mix with water using a hoe. Then take a trowel and place in the cracks. You may have to reform your steps, depending on where the disintegration is happening. If that is the case, you can use luan plywood and 2x4s. Then pour it into the areas. Using a wet trowel, you smooth when it's completed. Keep an eye on it to make sure there is no debris blown into the area until it is completely dry and set.

Remove all of the loose concrete. Paint the area with a concrete bonding liquid and then top with a concrete such as Quickcrete or Reddi-Mix which is made for just this purpose. Before it completely dries, sweep it lightly with a broom to give it a rough surface so that it is not slippery at any time. After it cures and winter sets in top the surface with a concrete protector and do this each and ever Fall so that you do not get the pitting from the salt. Purchase an ice melt that does not eat your concrete. There are some out there but you have to read the labels to be sure. I use the Alaskan Ice Melt.