How much did cutting unemployment benefits help the labor market?

Quite a bit.  There is a new NBER Working Paper on this topic by Hagedorn, Manovskii, and Mitman, showing (once again) that most supply curves slope upward, here is one key part from the abstract:

In levels, 1.8 million additional jobs were created in 2014 due to the benefit cut. Almost 1 million of these jobs were filled by workers from out of the labor force who would not have participated in the labor market had benefit extensions been reauthorized.

There is an ungated copy here (pdf).  Like the sequester, this is another area where the Keynesian analysts simply have not proven a good guide to understanding recent macroeconomic events.