Understanding CFLs in Electrical Circuits
A CFL broke a little towards the top when I touched it.


So, it's a mechanical failure of the glass tube? That means the near-vacuum is gone and the mercury filling now sticks to the electrodes.

The electrodes may still light up and also light the fluorescent powder at the ends of the glass tube. But they can't get the gas filling to conduct by itself anymore as the gas pressure is too high for that.

In low-pressure mercury gas, the electrons are accelerated by the electric field so much they can crack another electron (or more) from a mercury atom when they hit it. That results in an avalance effect, which enables current flow. At higher pressure, they hit the next atom before they are fast enough.